Vector Basics: Concepts, Operations, and Dot Product

Vectors are mathematical objects that have both magnitude and direction. They are essential in geometry, physics, and engineering, forming the foundation of spatial reasoning and coordinate calculations.

What Is a Vector?

Definition

A vector is a quantity that has both magnitude (size) and direction. It is represented as an arrow from a starting point $A$ to an endpoint $B$, written as $\vec{AB}$.

Equality of Vectors

Two vectors are equal if and only if they have the same magnitude and direction. Position does not matter.

Vector Operations

Step 1: Addition and Subtraction

In component form with $\vec{a} = (a_1, a_2)$ and $\vec{b} = (b_1, b_2)$:

$\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2)$

$\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2)$

Step 2: Scalar Multiplication

For a real number $k$:

$k\vec{a} = (ka_1, ka_2)$

• $k > 0$: same direction, magnitude scaled by $|k|$
• $k < 0$: opposite direction, magnitude scaled by $|k|$
• $k = 0$: zero vector $\vec{0}$

Step 3: Magnitude of a Vector

$|\vec{a}| = \sqrt{a_1^2 + a_2^2}$

A vector with magnitude 1 is called a unit vector.

The Dot Product

Definition

The dot product of $\vec{a}$ and $\vec{b}$ can be computed two ways:

Method 1 — Using the angle

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$

Method 2 — Using components

$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2$

Applications of the Dot Product

• Perpendicularity test: $\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}$
• Finding angles: $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$
• Projection: The projection of $\vec{b}$ onto $\vec{a}$ has length $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$

Practice Problems

Example 1: Dot Product Calculation

Problem: Given $\vec{a} = (3, 4)$ and $\vec{b} = (-1, 2)$, find $\vec{a} \cdot \vec{b}$ and the angle $\theta$ between them.

Example 2: Perpendicularity Condition

Problem: Find $k$ such that $\vec{a} = (2, k)$ and $\vec{b} = (3, -6)$ are perpendicular.

Top 3 Common Mistakes

1. Adding magnitudes directly — To find $|\vec{a} + \vec{b}|$, first add the components, then compute the magnitude. Never add $|\vec{a}| + |\vec{b}|$ directly.
2. Treating the dot product as a vector — $\vec{a} \cdot \vec{b}$ produces a scalar (number), not a vector.
3. Forgetting the zero vector — $\vec{0}$ has no defined direction. Check for zero vector exceptions in perpendicularity and parallelism problems.

Related Concepts

• Trigonometry — angle calculations
• Pythagorean Theorem — geometry foundation
• Limits — calculus connection