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Topic 3 of 20

Vectors

A vector is a quantity with both magnitude and direction. Forces, velocities, accelerations, and momenta are all vectors. Mastering vector decomposition into components — and the rules for adding and multiplying vectors — is essential for every topic that follows in physics.

Key Concepts

Scalar vs. Vector

A scalar has magnitude only (e.g., mass, temperature, speed). A vector has both magnitude and direction (e.g., displacement, velocity, force). Vectors are denoted

\vec{A}

or in bold.

Vector Components

Any 2D vector

\vec{A}

making angle

\theta

with the +x axis has components

A_x = A\cos\theta

and

A_y = A\sin\theta

, where

A = |\vec{A}|

is the magnitude.

Vector Addition

Add vectors component-wise: if

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

, then

C_x = A_x + B_x

and

C_y = A_y + B_y

. This is equivalent to the graphical head-to-tail method.

Unit Vectors

Unit vectors

\hat{i}

\hat{j}

\hat{k}

point along the positive x, y, z axes, each with magnitude 1. Any vector can be written

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

Dot Product (Scalar Product)

The dot product

\vec{A}\cdot\vec{B} = AB\cos\theta = A_xB_x + A_yB_y

gives a scalar. It is zero when vectors are perpendicular. It measures the projection of one vector onto another.

Cross Product (Vector Product)

The cross product

\vec{A}\times\vec{B}

gives a vector perpendicular to both, with magnitude

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

. Direction given by the right-hand rule. Used for torque and angular momentum.

Key Equations

Magnitude of a Vector

|\vec{A}| = A = \sqrt{A_x^2 + A_y^2}

Length of a 2D vector from its components via the Pythagorean theorem.

Direction Angle

\theta = \arctan\!\left(\frac{A_y}{A_x}\right)

Angle the vector makes with the positive x-axis. Use atan2 to get the correct quadrant.

Dot Product

\vec{A}\cdot\vec{B} = AB\cos\theta = A_xB_x + A_yB_y

Scalar result; equals zero for perpendicular vectors, maximum when parallel.

Cross Product Magnitude

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

The area of the parallelogram spanned by the two vectors.

Cross Product (component form)

\vec{A}\times\vec{B} = (A_yB_z - A_zB_y)\hat{i} - (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k}

Full 3D component expression for the cross product.

Worked Example

Adding Two Vectors

Problem

Vector $\vec{A}$ has magnitude 5 at 37° above the +x axis. Vector $\vec{B}$ has magnitude 8 pointing along the +x axis. Find the magnitude and direction of $\vec{A} + \vec{B}$ .

Solution

Decompose each vector into components:

A_x = 5\cos37° \approx 3.99, \quad A_y = 5\sin37° \approx 3.01

B_x = 8, \quad B_y = 0

Add components:

C_x = 3.99 + 8 = 11.99, \quad C_y = 3.01 + 0 = 3.01

Find magnitude and angle:

C = \sqrt{11.99^2 + 3.01^2} \approx 12.4

\theta = \arctan\!\left(\frac{3.01}{11.99}\right) \approx 14.1°

Answer |C| ≈ 12.4, directed 14.1° above the +x axis.

Practice

Exercises

7 problems

1 of 7

A vector has components $A_x = 3$ and $A_y = 4$ . What is its magnitude?

Your answer

(unitless)

2 of 7

A vector has magnitude $12$ and points at $60°$ above the $+x$ axis. What is its $x$ -component?

Your answer

(same as vector)

3 of 7

The same vector (magnitude $12$ , angle $60°$ ). What is its $y$ -component? Give your answer to one decimal place.

Your answer

(same as vector)

4 of 7

Vector $\vec{A} = (5,\,0)$ and $\vec{B} = (0,\,12)$ . What is $|\vec{A} + \vec{B}|$ ?

Your answer

(same as vector)

5 of 7

Two vectors each have magnitude $8$ . The angle between them is $60°$ . What is their dot product?

Your answer

(unitless)

6 of 7

Two vectors have magnitudes $5$ and $4$ . The angle between them is $30°$ . What is the magnitude of their cross product?

Your answer

(unitless)

7 of 7

A force of $10 \text{ N}$ is applied at $37°$ above the horizontal. What is the horizontal component of this force? (Use $\cos37° \approx 0.80$ )

Your answer

Key Takeaways

Always resolve vectors into x and y components before doing arithmetic — never add magnitudes directly.
The angle $\theta$ in component equations is measured from the positive x-axis; adjust signs for other quadrants.
Dot product gives a scalar and tells you how "parallel" two vectors are.
Cross product gives a vector perpendicular to both inputs and is used for torque and angular momentum.
The unit vectors $\hat{i}$ , $\hat{j}$ , $\hat{k}$ are orthonormal: $\hat{i}\cdot\hat{i}=1$ , $\hat{i}\cdot\hat{j}=0$ , $\hat{i}\times\hat{j}=\hat{k}$ .