Vector Operations

A vector is a mathematical quantity that has both magnitude (length) and direction. In a 2D coordinate plane, vectors are commonly written in component form as $\vec{v} = \langle a, b \rangle$, where $a$ represents the horizontal shift and $b$ represents the vertical shift.

Basic Vector Operations

When working with vectors in component form, operations are performed component by component.

• Addition: To add two vectors, add their corresponding components. $\langle u_1, u_2 \rangle + \langle v_1, v_2 \rangle = \langle u_1 + v_1, u_2 + v_2 \rangle$
• Subtraction: To subtract two vectors, subtract their corresponding components. $\langle u_1, u_2 \rangle - \langle v_1, v_2 \rangle = \langle u_1 - v_1, u_2 - v_2 \rangle$
• Scalar Multiplication: To multiply a vector by a real number (a scalar) $k$, multiply each component by $k$. This scales the vector's length and can reverse its direction if $k$ is negative. $k\langle a, b \rangle = \langle ka, kb \rangle$

Magnitude of a Vector

The magnitude (or length) of a vector $\vec{v} = \langle a, b \rangle$ is denoted by $|\vec{v}|$ or $||\vec{v}||$. It can be found using the Pythagorean theorem:

$|\vec{v}| = \sqrt{a^2 + b^2}$

Unit Vectors

A unit vector is a vector that has a magnitude of exactly $1$. To find a unit vector $\hat{u}$ that points in the exact same direction as a given vector $\vec{v}$, you divide the vector by its own magnitude:

$\hat{u} = \frac{1}{|\vec{v}|} \vec{v} = \left\langle \frac{a}{|\vec{v}|}, \frac{b}{|\vec{v}|} \right\rangle$

Example Problems

Example 1: Given $\vec{u} = \langle 3, 4 \rangle$ and $\vec{v} = \langle -1, 2 \rangle$, find $2\vec{u} - 3\vec{v}$ and $|2\vec{u} - 3\vec{v}|$.

1. First, find scalar multiples: $2\vec{u} = 2\langle 3, 4 \rangle = \langle 6, 8 \rangle$ $3\vec{v} = 3\langle -1, 2 \rangle = \langle -3, 6 \rangle$
2. Next, subtract the scaled vectors: $2\vec{u} - 3\vec{v} = \langle 6 - (-3), 8 - 6 \rangle = \langle 9, 2 \rangle$
3. Finally, find the magnitude of the resulting vector: $|2\vec{u} - 3\vec{v}| = \sqrt{9^2 + 2^2} = \sqrt{81 + 4} = \sqrt{85}$

Example 2: Find the unit vector in the direction of $\langle -5, 12 \rangle$.

1. Let $\vec{v} = \langle -5, 12 \rangle$. First, calculate its magnitude: $|\vec{v}| = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
2. Divide the vector by its magnitude to get the unit vector: $\hat{v} = \frac{1}{13} \langle -5, 12 \rangle = \left\langle -\frac{5}{13}, \frac{12}{13} \right\rangle$