Vector Operations: Addition, Dot Product & Cross Product

Why Study Vectors?

Vectors are quantities with both magnitude and direction. They are the mathematical language of forces in physics, directions in navigation, and coordinates in computer graphics. Mastering vector operations is essential for any STEM discipline.

Learning Goals: By the end of this guide, you should be able to:

1. Add, subtract and scale vectors algebraically and geometrically.
2. Compute the dot product and use it to find angles.
3. Compute the cross product and use it to find areas and normals.
4. Apply vector operations to solve geometry and physics problems.

Vector Basics

A vector in 2D or 3D is written as $\vec{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ or $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$.

Magnitude

Unit Vector

Addition and Scalar Multiplication

Addition: $\vec{a} + \vec{b} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ a_3 + b_3 \end{pmatrix}$

Geometrically: place $\vec{b}$ at the tip of $\vec{a}$ (head-to-tail rule).

Scalar multiplication: $k\vec{a} = \begin{pmatrix} ka_1 \\ ka_2 \\ ka_3 \end{pmatrix}$

• $k > 1$: stretches the vector
• $0 < k < 1$: shrinks the vector
• $k < 0$: reverses direction

Dot Product (Scalar Product)

Key Properties

Finding the Angle

Cross Product (Vector Product)

The cross product is defined only in 3D:

Key Properties

• Result is a vector perpendicular to both $\vec{a}$ and $\vec{b}$.
• Magnitude: $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ = area of the parallelogram formed by $\vec{a}$ and $\vec{b}$.
• Direction: given by the right-hand rule.
• Anti-commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$

Worked Examples

Example 1: Angle Between Vectors

Find the angle between $\vec{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}$.

Example 2: Area of a Triangle

Find the area of the triangle with vertices $A(1,0,0)$, $B(0,1,0)$, $C(0,0,1)$.

Common Mistakes

1. Confusing dot product and cross product — Dot product gives a scalar; cross product gives a vector. Check which one the question asks for.
2. Forgetting that cross product is anti-commutative — $\vec{a} \times \vec{b} \neq \vec{b} \times \vec{a}$. The order matters.
3. Errors in the determinant expansion — Be careful with signs: the $\hat{j}$ component has a negative sign in the cofactor expansion.

Exam Tips (A-Level / AP / IB)

• To check perpendicularity, compute the dot product. If it's zero, the vectors are perpendicular.
• The cross product is the fastest way to find a normal vector to a plane defined by two direction vectors.
• For projection of $\vec{a}$ onto $\vec{b}$: $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$.

Frequently Asked Questions

What is the difference between a scalar and a vector?

A scalar has only magnitude (e.g., temperature, mass). A vector has both magnitude and direction (e.g., velocity, force).

Does the cross product exist in 2D?

Not exactly. In 2D, the "cross product" of $\vec{a} = (a_1, a_2)$ and $\vec{b} = (b_1, b_2)$ gives a single scalar $a_1 b_2 - a_2 b_1$, which equals the signed area of the parallelogram.

Related Topics

• Spatial Geometry — Vectors provide the algebraic toolkit for 3D geometry.
• Complex Numbers — Complex number multiplication has a vector rotation interpretation.
• Trigonometric Graphs — Dot product formula involves cosine; cross product involves sine.