Vectors

Why Numbers Alone Are Not Enough

Consider a simple question: if a force of 10 N and a force of 8 N act on a block, what is the total resulting force?

Treated as ordinary numbers, the two forces seem to add to 18 N. In fact, that is only one of many possibilities. What if the 10 N force is pushing the block forward, and the 8 N force is pushing it backward? The net effect would be a mere 2 N forward. What if they pull at right angles, or at some other diagonal angle to each other? The answer changes completely. You simply cannot calculate the total effect without knowing their directions.

Numbers alone, therefore, are not always enough to describe nature. Some quantities require both a magnitude (how much) and a direction (which way) to make sense. This chapter develops the tools used to describe and manipulate such quantities.

Classifying Our Measurements

In physics, we describe the universe by measuring its properties. Any property of an object or system that can be measured and quantified is called a physical quantity. For instance, we can measure how heavy a stone is (mass), how hot a cup of tea is (temperature), or how fast a runner is moving (speed).

To handle these measurements mathematically, we group physical quantities into two fundamental classes based on whether they care about spatial direction: scalars and vectors.

Scalar Quantities

A scalar quantity is one that is completely described by a single numerical value—its magnitude—accompanied by its appropriate physical unit. Scalars are simple: they tell us "how much" or "how many," and they do not have any direction in space.

Consider these everyday examples of scalars:

• Mass: An apple has a mass of $150 \text{ g}$. It makes no sense to ask "in what direction" the mass points.
• Temperature: A bowl of soup is at $60^\circ\text{C}$. It is hot in all directions equally.
• Distance: A walk along a winding path may cover a distance of $5 \text{ km}$. This is the total length of the path you walked, regardless of where it led.
• Speed: A car's speedometer reads $60 \text{ km/h}$. It tells you how fast the car is moving, but not where it is heading.
• Energy: A battery stores $10 \text{ Wh}$ of energy. Energy is a capacity, not a path.

Because scalars are just numbers, you can add, subtract, multiply, and divide them using the ordinary rules of algebra.

Vector Quantities

A vector quantity is one that requires both a magnitude and a direction in space for its complete description. If you omit the direction, the description is physically incomplete.

The following examples show how vectors differ from scalars:

• Displacement: Instead of just distance, displacement tells you how your position changed. If you walk $5 \text{ km}$ East, you have specified a vector.
• Velocity: Velocity is speed with a direction. A car traveling at $60 \text{ km/h}$ North has a velocity vector.
• Force: Pushing a box with $20 \text{ N}$ of force downwards has a very different physical outcome than pushing it $20 \text{ N}$ upwards.
• Acceleration: Gravity accelerates falling objects at $9.8 \text{ m/s}^2$ straight down.

Because vectors carry spatial direction, we cannot add them using standard arithmetic. A displacement of $3 \text{ m}$ plus a displacement of $4 \text{ m}$ does not necessarily equal $7 \text{ m}$—it depends entirely on how the two movements align. The rules for combining vectors are developed in the sections that follow.

Two Ways to Represent a Vector

To work with vectors, we need a clear system to talk about them (notation) and visualize them (diagrams). In physics, we rely on two main methods of representation: a graphical representation (drawing arrows) and an algebraic representation (using symbols and numbers). We switch between these two modes constantly, using diagrams to build intuition and algebra to perform exact calculations.

Graphical Representation

The most intuitive way to represent a vector is diagrammatically as an arrow in space. This arrow is a literal model of the vector's two properties:

• The length of the arrow is drawn to scale to represent the magnitude of the vector. A longer arrow means a larger quantity (e.g., a stronger force or faster speed).
• The direction the arrowhead points is the direction of the vector.

The starting point of the arrow is called the tail, and the pointed end is the head. In diagrams, the angle of the arrow is typically measured from a reference direction, such as the positive x-axis or North.

Symbolic Notation

When writing equations, how do we tell the reader that a letter represents a vector rather than a normal number? We use special typographical symbols:

• The Arrow Notation: We write a small arrow above the variable, such as $\vec{A}$ or $\vec{F}$. The arrow marks the symbol as a vector: a reminder that it cannot be combined with other terms by ordinary arithmetic.
• Boldface Notation: In printed textbooks, vectors are often printed in bold, like A or F. Because boldface is difficult to write by hand, arrow notation is preferred for handwritten work.

The magnitude of a vector is a scalar—it is simply a positive number with a unit. We write the magnitude of vector $\vec{A}$ using absolute value bars as $|\vec{A}|$, or simply as the letter in standard italics, $A$.

For instance, if $\vec{v}$ represents a velocity vector of $10 \text{ m/s}$ East, its magnitude is $v = |\vec{v}| = 10 \text{ m/s}$.

Analytical Representation: Component Form

Drawing arrows helps us visualize physical situations, but it is not convenient for precise mathematics. Instead of drawing, we can represent a vector analytically by breaking it down into components along perpendicular coordinate axes (usually the horizontal x-axis and vertical y-axis).

You can think of a component as the "projection" or shadow of the vector along a specific direction. To write vectors in this component form, we introduce unit vectors, which are vectors of magnitude 1 pointing along the coordinate axes:

• $\hat{i}$ (pronounced "i-hat") points in the positive x-direction.
• $\hat{j}$ (pronounced "j-hat") points in the positive y-direction.
• $\hat{k}$ (pronounced "k-hat") points in the positive z-direction (for three dimensions).

Using these unit vectors as direction markers, we write any vector $\vec{A}$ as a sum of its horizontal component $A_x$ and its vertical component $A_y$:

$$\vec{A} = A_x\hat{i} + A_y\hat{j}$$

For example, if a force pulls 3 Newtons to the right and 4 Newtons upward, we write it as $\vec{F} = 3\hat{i} + 4\hat{j} \text{ N}$. This is the analytical component form of the vector.

Component Form in Use: Two Walks

Take $\hat{i}$ to point East, $\hat{j}$ North, and $\hat{k}$ vertically up.

A walk in the plane. A person walks $30 \text{ m}$ East and then $40 \text{ m}$ North. The two legs are $30\hat{i}$ and $40\hat{j}$, so the total displacement is

$$\vec{d} = 30\hat{i} + 40\hat{j} \text{ m}, \qquad d = \sqrt{30^2 + 40^2} = 50 \text{ m}$$

A walk that leaves the plane. Another person walks $20 \text{ m}$ West, then $10 \text{ m}$ North, then enters a lift and rises $15 \text{ m}$. West is the negative $\hat{i}$ direction and the lift ride adds a $\hat{k}$ component:

$$\vec{d} = -20\hat{i} + 10\hat{j} + 15\hat{k} \text{ m}, \qquad d = \sqrt{(-20)^2 + 10^2 + 15^2} = \sqrt{725} \approx 26.9 \text{ m}$$

Each movement simply contributes its own component, and the components never mix — that is what makes the component form the working language of vector calculations.

The Position Vector

One particular vector deserves its own name because it appears everywhere in mechanics: the position vector. The position vector of a point is the displacement vector starting from the origin and ending at that point.

Because it is measured from the origin, a position vector depends entirely on the choice of origin and the orientation of the coordinate axes you are using — change either one, and the position vector of the same physical point changes too. This is different from a general vector like a force or a velocity, whose value doesn't care where you place the origin.

The Cartesian System of Vectors in 3D

So far we have written vectors using two perpendicular unit vectors, $\hat{i}$ and $\hat{j}$, for problems confined to a flat plane. But most of the real world is three-dimensional, so we extend the same idea by introducing a third unit vector, $\hat{k}$, pointing along the z-axis. Together, $\hat{i}$, $\hat{j}$, and $\hat{k}$ are three mutually perpendicular unit vectors along the x, y, and z axes of a right-handed coordinate system.

The three mutually perpendicular unit vectors i-hat, j-hat, and k-hat along the x, y, and z axes of a right-handed coordinate system; drag the scene to turn it, and zoom with the buttons

The Position Vector in 3D

Consider a point $P(a, b, c)$ in three-dimensional space. The position vector $\vec{r}$ of $P$, drawn from the origin $O$, is written as the sum of its three components along the axes:

$$\overrightarrow{OP} = \vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$$

The position vector r of point P(a,b,c) built from its components: a along the x axis, then b parallel to the y axis, then c straight up. Drag the scene to turn it, and zoom with the buttons.

The magnitude of this position vector follows directly from the three-dimensional Pythagorean theorem:

$$r = \sqrt{a^2 + b^2 + c^2}$$

Equality of Vectors in Component Form

Writing vectors in Cartesian form also gives us a clean, purely algebraic test for equality: two vectors expressed in $\hat{i}, \hat{j}, \hat{k}$ form are equal only when their corresponding components are separately equal. That is, if $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$, then $\vec{A} = \vec{B}$ if and only if $A_x = B_x$, $A_y = B_y$, and $A_z = B_z$ all hold simultaneously.

Properties of Vectors

Just as numbers have rules and properties (like the identity property $x + 0 = x$), vectors have their own unique properties that govern how they behave. Before we perform algebra on vectors, we must establish these fundamental properties.

1. Equality of Vectors

Two vectors, $\vec{A}$ and $\vec{B}$, are defined as equal if, and only if, they have the same magnitude and the same direction. We write this as $\vec{A} = \vec{B}$.

Notice what is missing from this definition: position. It does not matter where a vector starts or ends in space. As long as its length is the same and it points in the same direction, it is the same vector. This means vectors are portable—you can slide them anywhere in space without altering their value, which is a powerful shortcut when adding vectors graphically.

2. The Unit Vector

A unit vector is a special vector with a magnitude of exactly 1. It carries no physical units of its own; its sole purpose is to point, establishing a direction in space. It is a mathematical signpost.

If you have any vector $\vec{A}$, you can find its corresponding unit vector, written with a caret or "hat" as $\hat{A}$ (pronounced "A-hat"), by dividing the vector by its own magnitude:

$$\hat{A} = \frac{\vec{A}}{|\vec{A}|}$$

This operation scales the vector's length down to exactly 1 while preserving its direction. Consequently, any vector can be written as the product of its size (magnitude) and its direction (unit vector):

$$\vec{A} = A \hat{A}$$

3. The Negative of a Vector

The negative of a vector $\vec{A}$, written as $-\vec{A}$, is a vector with the same magnitude (length) as $\vec{A}$, but pointing in the exact opposite direction (a $180^\circ$ rotation).

4. The Zero (Null) Vector

The zero vector, denoted as $\vec{0}$, is a vector with a magnitude of exactly zero. Because it has no length, its direction is undefined. It acts as the identity element for vector addition, meaning $\vec{A} + \vec{0} = \vec{A}$.

A zero vector arises naturally in physics when physical effects cancel out (e.g., when equal forces pull in opposite directions) or when an object is at rest, resulting in a velocity of $\vec{0}$.

5. Parallel and Antiparallel Vectors

Vectors are described as parallel if they point in the exact same direction, regardless of whether their magnitudes are equal. The angle between parallel vectors is $0^\circ$.

Vectors are described as antiparallel if they point in exact opposite directions. The angle between antiparallel vectors is $180^\circ$.

6. The Angle Between Two Vectors

We often need to speak precisely about how two vectors are oriented relative to each other. The angle between two vectors $\vec{A}$ and $\vec{B}$ is defined as the angle through which one vector must be rotated about their common tail so that its direction coincides with the direction of the other vector.

To measure this angle correctly, always slide the two vectors (recall that vectors are portable) so that their tails meet at a common point, and then read off the angle between them at that point — never the angle measured between their arrowheads or along some other path.

Two Descriptions of the Same Vector

We have seen that a vector can be described in two ways: by its magnitude and direction (like a velocity of $10 \text{ m/s}$ at $30^\circ$ above the horizontal) or by its components (like $8.66\hat{i} + 5\hat{j} \text{ m/s}$).

To solve physics problems, we must translate between these two descriptions. This process is called vector resolution. In this section, we will learn how to break a diagonal vector down into its perpendicular parts, and conversely, how to combine those parts back into a single magnitude and direction.

Resolving a Vector into Components

Consider a vector $\vec{A}$ making an angle $\theta$ with the positive x-axis. By dropping perpendicular lines from the head of the vector to the axes, we form a right-angled triangle where the vector is the hypotenuse and the components are the legs.

From Components Back to Magnitude and Direction

What if you know the components $A_x$ and $A_y$ and want to find the overall vector's length and heading? We can reconstruct the vector using the same right triangle.

1. Calculating the Magnitude

Because the components $A_x$ and $A_y$ form the perpendicular legs of a right triangle, the magnitude $A$ (the hypotenuse) is found using the Pythagorean theorem:

$$A = \sqrt{A_x^2 + A_y^2}$$

Since squaring negative numbers yields positive results, the magnitude is always a positive number (or zero), representing the absolute length of the vector.

2. Calculating the Direction (with Quadrant Correction)

The direction angle $\theta$ is related to the components by the tangent function, which is the ratio of the opposite leg to the adjacent leg:

$$\tan\theta = \frac{A_y}{A_x}$$

To solve for $\theta$, we use the inverse tangent function:

$$\theta = \tan^{-1}\left( \frac{A_y}{A_x} \right)$$

Combining Vectors: The Net Effect

Suppose two persons push a box, each with a force of $10 \text{ N}$. What is the total force on the box? The question cannot be answered as it stands. If both push the same way, the total is $20 \text{ N}$. If they push in opposite directions, the pushes cancel and the total is zero. If they push at right angles, the total is $\sqrt{10^2 + 10^2} \approx 14.1 \text{ N}$, directed between them. Without the directions, the sum of two forces is simply not determined.

Displacements show the same behaviour, and are the easiest vectors to picture. Walk $3 \text{ m}$ and then $4 \text{ m}$: you end $7 \text{ m}$ from the start only if both walks point the same way. Walk them at right angles and you end $\sqrt{3^2 + 4^2} = 5 \text{ m}$ from the start; walk the second leg straight back and you end only $1 \text{ m}$ away. The order and directions of the two legs decide where you finish.

Adding vectors therefore needs its own rules. The sections that follow build those rules: the Triangle Law and the Parallelogram Law add vectors by drawing, and the component method adds them by calculation.

The Resultant Vector

The sum of two or more vectors is called the resultant vector (often denoted as $\vec{R}$). The resultant is a single vector that produces the same net effect as all the individual vectors combined. For example, the resultant force on an object determines its acceleration.

$$\vec{R} = \vec{A} + \vec{B}$$

Example: A Tug-of-War

Consider a tug-of-war. Team A pulls left with 100 N ($\vec{F}_A$) and Team B pulls right with 80 N ($\vec{F}_B$). The resultant force is not 180 N. Since they are in opposite directions, the net effect is a force of 20 N to the left. So, $\vec{R} = \vec{F}_A + \vec{F}_B = 20$ N to the left.

Range of Possible Resultants

A fascinating aspect of vector addition is that two vectors with fixed magnitudes can produce a resultant with a wide range of possible magnitudes, depending on the angle between them.

Suppose we have two vectors, $\vec{A}$ with magnitude $A$ and $\vec{B}$ with magnitude $B$. The magnitude of their resultant, $R$, will be:

• Maximum when $\vec{A}$ and $\vec{B}$ are parallel (angle = 0°). Then, $R_{max} = A + B$.
• Minimum when $\vec{A}$ and $\vec{B}$ are antiparallel (angle = 180°). Then, $R_{min} = |A - B|$.

For any other angle between them, the resultant's magnitude $R$ will be somewhere in between these two extremes: $|A - B| \le R \le A + B$.

Methods of Addition

So, how do we actually perform this addition, especially when the vectors are not parallel or antiparallel? We will learn two primary approaches:

1. Graphical Methods: Using geometric constructions like the 'Triangle Law' and 'Parallelogram Law' to visualize the resultant.
2. Analytical Method: Using vector components and trigonometry for precise calculations.

We will explore these methods in detail in the following sections.

The Head-to-Tail Method

The Triangle Law provides our first graphical method for adding two vectors. It's an intuitive approach based on a 'journey' metaphor. If you walk along vector $\vec{A}$ and then immediately walk along vector $\vec{B}$, where do you end up? The resultant vector, $\vec{R}$, is the single vector that takes you directly from your starting point to your final destination.

This method is formally called the head-to-tail method because of how we arrange the vectors.

Statement of the Triangle Law

If two vectors are represented in magnitude and direction by two sides of a triangle taken in the same order (head-to-tail), then their resultant is represented in magnitude and direction by the third side of the triangle, taken in the opposite order (from the tail of the first to the head of the second).

Step-by-Step Graphical Construction:

1. Draw the First Vector: Draw vector $\vec{A}$ to a chosen scale and in its specified direction.
2. Draw the Second Vector: Place the tail of vector $\vec{B}$ at the head of vector $\vec{A}$. Draw $\vec{B}$ to the same scale and in its correct direction.
3. Draw the Resultant: The resultant vector $\vec{R} = \vec{A} + \vec{B}$ is the vector drawn from the tail of $\vec{A}$ to the head of $\vec{B}$.
4. Measure the Resultant: Measure the length of $\vec{R}$ and use the scale to find its magnitude. Measure its angle to find its direction.

Vector Addition is Commutative

The order in which two vectors are added does not change the result. Suppose we add $\vec{B} + \vec{A}$? We would first draw $\vec{B}$, then draw $\vec{A}$ starting from the head of $\vec{B}$. As you can see from the diagram, we arrive at the exact same resultant vector $\vec{R}$.

$$\vec{A} + \vec{B} = \vec{B} + \vec{A}$$

This shows that vector addition is commutative, just like the addition of scalar numbers.

Vector Addition is Associative

When we add three or more vectors, does the way we group them matter? Suppose we want $\vec{A} + \vec{B} + \vec{C}$. We could first add $\vec{A}$ and $\vec{B}$, then add $\vec{C}$ to that sum; or we could first add $\vec{B}$ and $\vec{C}$, then add $\vec{A}$ to that sum. If you draw both groupings head-to-tail, you'll find they produce the exact same final resultant vector, closing the same overall polygon. This confirms that vector addition is associative:

$$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$$

This means that when adding several vectors, you're free to group and reorder them however is most convenient, and the final resultant will never change.

Magnitude and Direction of the Resultant

The triangle construction also gives an exact formula for the resultant. Let $\vec{B}$ be drawn from the head of $\vec{A}$, leaving it at the angle $\alpha$ to the direction of $\vec{A}$. Extend the line of $\vec{A}$ and drop a perpendicular from the head of $\vec{B}$ onto it. The perpendicular has length $B\sin\alpha$, and it meets the extension at a distance $B\cos\alpha$ past the head of $\vec{A}$.

Extending to Many Vectors: The Polygon Law

The head-to-tail idea behind the Triangle Law extends naturally to more than two vectors. To add several vectors, draw them one after another in order, always starting the next vector from the head of the previous one. The resultant is then the single vector drawn from the tail of the very first vector to the head of the very last vector — closing the polygon formed by all the vectors. This generalization is called the Polygon Law of Vector Addition.

A Useful Corollary: Closed Polygons Sum to Zero

An important special case falls directly out of the Polygon Law: if a set of $n$ vectors, added head-to-tail in this way, forms a closed polygon — meaning the head of the last vector lands exactly back on the tail of the first — then their resultant is the zero vector. There is no 'gap' left to close, so there is nothing left for the resultant to represent.

The Tail-to-Tail Method

The Parallelogram Law is another graphical method for adding two vectors. While the Triangle Law is a 'head-to-tail' method, the Parallelogram Law is a 'tail-to-tail' method. It is particularly useful for visualizing problems where two vectors act on the same point, such as two forces pulling on an object.

Statement of the Parallelogram Law

If two vectors, acting simultaneously at a point, are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from that point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that same point.

Step-by-Step Graphical Construction:

1. Draw Vectors Tail-to-Tail: Draw vector $\vec{A}$ and vector $\vec{B}$ starting from the same origin point.
2. Complete the Parallelogram: Draw a line parallel to $\vec{A}$ from the head of $\vec{B}$, and another line parallel to $\vec{B}$ from the head of $\vec{A}$. These lines will intersect to form a parallelogram.
3. Draw the Resultant: The resultant vector $\vec{R} = \vec{A} + \vec{B}$ is the diagonal of the parallelogram that starts from the common origin point.

Connection to the Triangle Law

The Parallelogram Law is just a different way to visualize the Triangle Law. Notice that the side of the parallelogram opposite to $\vec{B}$ is equal to $\vec{B}$. If you hide the original $\vec{B}$ and the side parallel to $\vec{A}$, you are left with a perfect 'head-to-tail' addition of $\vec{A}$ and $\vec{B}$ forming a triangle with the resultant $\vec{R}$.

The Same Resultant, Seen on the Parallelogram

The formula for the resultant was derived in the previous section from the triangle construction:

$$R = \sqrt{A^2 + B^2 + 2AB\cos\alpha} \qquad \tan\theta = \frac{B\sin\alpha}{A + B\cos\alpha}$$

The parallelogram shows exactly the same right triangle. Draw $\vec{A}$ and $\vec{B}$ tail-to-tail with the angle $\alpha$ between them and complete the parallelogram; the diagonal from the common tail is $\vec{R}$. Dropping a perpendicular from the head of $\vec{R}$ onto the extension of $\vec{A}$ reproduces the base $A + B\cos\alpha$ and the height $B\sin\alpha$, because the far side of the parallelogram is just $\vec{B}$ shifted to the head of $\vec{A}$.

The Meaning of Vector Subtraction

Vector subtraction is not a new operation, but rather a special case of addition. The key insight is this: to subtract a vector, you add its negative.

So, the operation $\vec{A} - \vec{B}$ is defined as:

$$\vec{R} = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$$

This simple definition allows us to use all the tools we've already learned for vector addition (Triangle Law, Parallelogram Law, Component Method) to perform subtraction.

Graphical Subtraction

To subtract $\vec{B}$ from $\vec{A}$ graphically, we first need to find the negative of $\vec{B}$, which we call $-\vec{B}$. This is a vector with the same length as $\vec{B}$ but pointing in the exact opposite direction. Then, we add $\vec{A}$ and $-\vec{B}$ using the head-to-tail method.

Subtraction by Components

The component method makes subtraction incredibly straightforward. To find the components of the resultant $\vec{R} = \vec{A} - \vec{B}$, you simply subtract the components of $\vec{B}$ from the components of $\vec{A}$:

$$R_x = A_x - B_x$$
$$R_y = A_y - B_y$$

Once you have $R_x$ and $R_y$, you can find the magnitude and direction of the resultant vector $\vec{R}$ just as you did with addition.

Example: Given $\vec{A} = 5\hat{i} + 2\hat{j}$ and $\vec{B} = 3\hat{i} + 4\hat{j}$. Find $\vec{A} - \vec{B}$.

$R_x = 5 - 3 = 2$
$R_y = 2 - 4 = -2$

So, $\vec{R} = 2\hat{i} - 2\hat{j}$.

Vector Subtraction is Not Commutative

We know vector addition is commutative, but subtraction is a different story. Consider $\vec{A} - \vec{B}$ and $\vec{B} - \vec{A}$. Since $\vec{B} - \vec{A} = -(\vec{A} - \vec{B})$, these two vectors are negatives of each other — same length, opposite direction. This means their magnitudes are equal:

$$|\vec{A} - \vec{B}| = |\vec{B} - \vec{A}|$$

but the vectors themselves are certainly not equal, since a vector and its negative point in opposite directions:

$$\vec{A} - \vec{B} = -(\vec{B} - \vec{A}) \quad \Rightarrow \quad \vec{A} - \vec{B} \neq \vec{B} - \vec{A}$$

So vector subtraction is non-commutative: swapping the order flips the direction of the result even though it leaves the size unchanged. It's easy to mistake the equal-magnitude fact for full equality — don't make that error.

Subtraction as a Diagonal of the Parallelogram

Recall that when $\vec{A}$ and $\vec{B}$ are drawn tail-to-tail, the parallelogram they form has two diagonals. The diagonal starting from the common tail represents the sum $\vec{A} + \vec{B}$ (the Parallelogram Law of addition). The other diagonal — the one connecting the two heads, which does not pass through the common tail — represents the subtraction: it is $\vec{A} - \vec{B}$ when drawn pointing from the head of $\vec{B}$ toward the head of $\vec{A}$, or $\vec{B} - \vec{A}$ when drawn in the opposite sense, from the head of $\vec{A}$ toward the head of $\vec{B}$. This single diagonal, read in either direction, geometrically encodes both non-commutative results at once.

The Physical Meaning of Subtraction

Vector subtraction is not just a mathematical curiosity; it's essential for describing important physical concepts. The most common use is to find the change in a vector quantity.

For example, change in velocity ($\Delta \vec{v}$) is defined as the final velocity minus the initial velocity:

$$\Delta \vec{v} = \vec{v}_{final} - \vec{v}_{initial}$$

This $\Delta \vec{v}$ vector is crucial for understanding acceleration. A car turning a corner at a constant speed still has a change in velocity (and thus accelerates) because its direction vector has changed, and $\vec{v}_{final} - \vec{v}_{initial}$ is not zero.

The Analytical Method

Graphical methods like the Triangle and Parallelogram laws are excellent for visualizing vector addition. However, they rely on precise drawing and measurement, which can be cumbersome and inaccurate. For precise and complex problems (especially with more than two vectors), we need a more robust technique: the analytical method of addition by components.

The core idea is simple but powerful: instead of struggling with adding slanted arrows, we break each vector down into its simple, perpendicular components (x and y). We then add all the x-components together and all the y-components together, which is easy because they lie on the same line. Finally, we reassemble the resultant vector from its summed components.

The Four-Step Process

Adding vectors $\vec{A}$, $\vec{B}$, $\vec{C}$, ... to find the resultant $\vec{R} = \vec{A} + \vec{B} + \vec{C} + ...$ involves four clear steps.

Worked Example

We add two vectors: $\vec{A}$ = 5.0 m at 37° and $\vec{B}$ = 3.0 m at 150°.

Step 1: Components
$A_x = 5.0 \cos(37°) = 5.0 \times 0.8 = 4.0$ m
$A_y = 5.0 \sin(37°) = 5.0 \times 0.6 = 3.0$ m
$B_x = 3.0 \cos(150°) = 3.0 \times (-0.866) = -2.6$ m
$B_y = 3.0 \sin(150°) = 3.0 \times 0.5 = 1.5$ m

Step 2: Sum
$R_x = 4.0 + (-2.6) = 1.4$ m
$R_y = 3.0 + 1.5 = 4.5$ m

Step 3: Magnitude
$R = \sqrt{(1.4)^2 + (4.5)^2} = \sqrt{1.96 + 20.25} = \sqrt{22.21} \approx 4.7$ m

Step 4: Direction
$\theta_R = \tan^{-1}\left(\frac{4.5}{1.4}\right) = \tan^{-1}(3.21) \approx 72.7°$
(Since $R_x$ and $R_y$ are both positive, the resultant is in the first quadrant and the angle is correct.)

Multiplying Vectors

We've mastered adding and subtracting vectors. Now, we venture into multiplication. With scalars, multiplication is simple: 3 times 4 is always 12. But with vectors, it's more complex. Why? Because we have to consider the directions.

In physics, we often need to combine two vectors to produce a third quantity. Sometimes, this new quantity is a scalar (like 'work'), and other times it's a new vector (like 'torque'). To handle these different physical situations, we need two distinct ways to multiply vectors.

The Two Products of Vectors

The two types of vector multiplication are defined by the kind of quantity they produce:

1. The Scalar Product (or Dot Product):
   This method multiplies two vectors to produce a scalar quantity. It is a measure of how much one vector lies along the direction of another. A classic example is the calculation of Work, where $W = \vec{F} \cdot \vec{d}$. The result, work, is a scalar (energy).
   Notation: $\vec{A} \cdot \vec{B}$
2. The Vector Product (or Cross Product):
   This method multiplies two vectors to produce a new vector. The resulting vector is perpendicular to the plane containing the original two vectors. A prime example is Torque, $\vec{\tau} = \vec{r} \times \vec{F}$, where the resulting torque vector's direction determines the axis of rotation.
   Notation: $\vec{A} \times \vec{B}$

The Defining Formulas

Here are the fundamental formulas for the two types of products. We will explore them in detail in the upcoming sections.

Scalar (Dot) Product:

The result is a scalar whose value is given by:

$$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta$$

where $\theta$ is the angle between the two vectors.

Vector (Cross) Product:

The result is a vector. Its magnitude is given by:

$$|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta$$

The direction of the resultant vector is given by the Right-Hand Rule, which we will learn about soon.

The Scalar Product

The dot product is a way of multiplying two vectors that tells you how much one vector 'goes along' the other. It measures the extent to which two vectors are aligned. The result is a single number, a scalar, hence the alternative name 'scalar product'.

Geometric Definition

The dot product of $\vec{A}$ and $\vec{B}$ is defined as the magnitude of $\vec{A}$ times the magnitude of $\vec{B}$ times the cosine of the angle $\theta$ between them.

$$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta$$

You can think of the term $|\vec{B}| \cos\theta$ as the 'shadow' or projection of vector $\vec{B}$ onto the direction of vector $\vec{A}$. So, the dot product is literally the magnitude of $\vec{A}$ multiplied by the component of $\vec{B}$ that is parallel to $\vec{A}$.

Calculating with Components

Start with the simplest dot products of all: the unit vectors themselves. Using $\vec{A}\cdot\vec{B} = AB\cos\theta$ with magnitudes of 1:

• What is $\hat{i}\cdot\hat{i}$? The angle between a vector and itself is $0^\circ$, so $\hat{i}\cdot\hat{i} = (1)(1)\cos 0^\circ = 1$. Likewise $\hat{j}\cdot\hat{j} = 1$ and $\hat{k}\cdot\hat{k} = 1$.
• What is $\hat{i}\cdot\hat{j}$? The axes are perpendicular, so $\hat{i}\cdot\hat{j} = (1)(1)\cos 90^\circ = 0$. Likewise $\hat{j}\cdot\hat{k} = 0$ and $\hat{k}\cdot\hat{i} = 0$.

So: a unit vector dotted with itself gives 1; two different unit vectors give 0.

Now compute a full dot product with the distributive law, multiplying term by term exactly as in algebra:

$$(3\hat{i} + 4\hat{j}) \cdot (2\hat{i} + 4\hat{j}) = 6\,(\hat{i}\cdot\hat{i}) + 12\,(\hat{i}\cdot\hat{j}) + 8\,(\hat{j}\cdot\hat{i}) + 16\,(\hat{j}\cdot\hat{j})$$

$$= 6(1) + 12(0) + 8(0) + 16(1) = 22$$

Every cross term dies, and only the $\hat{i}$-with-$\hat{i}$ and $\hat{j}$-with-$\hat{j}$ products survive. There is therefore no need to expand next time — multiply matching components and add:

$$\vec{A}\cdot\vec{B} = A_x B_x + A_y B_y + A_z B_z$$

For the vectors above: $(3)(2) + (4)(4) = 22$, the same answer in one line.

Key Properties and Applications

• Commutative: The order doesn't matter. $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$.
• Distributive over Addition: The dot product distributes across a sum of vectors just like ordinary multiplication distributes over addition: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$. This property is what lets us expand dot products of vector sums term by term.
• Finding the Angle Between Vectors: By rearranging the definition, we can find the angle between two vectors: $\cos\theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$.
• Checking for Perpendicularity: This is a crucial application. If two non-zero vectors are perpendicular (orthogonal), the angle between them is 90°, and $\cos(90°) = 0$. Therefore, the dot product of two perpendicular vectors is always zero.