1. Understanding Vector Components

A vector $\vec{v}$ represents a quantity with both magnitude (size) and direction. We describe it using "components" $\langle x, y \rangle$ which tell us how far to move horizontally and vertically.

The Component Rule:

If a vector starts at $(x_1, y_1)$ and ends at $(x_2, y_2)$:

$\vec{v} = \langle x_2 - x_1, y_2 - y_1 \rangle$

(1, 3) v

Part 1: Magnitude & Direction

Problem 1: Vector from (0,0) to (1,3)

Component Form

$\vec{v} = \langle 1, 3 \rangle$

Magnitude

$||\vec{v}|| = \sqrt{1^2 + 3^2} = \sqrt{10}$

Direction ($\theta$)

$\theta_v = \tan^{-1}(\frac{3}{1}) \approx 71.57^\circ$

Problem 2: Vector from (-1,-1) to (3,5)

Note: This vector does not start at the origin!

1

Subtract: $\vec{v} = \langle 3 - (-1), 5 - (-1) \rangle = \langle 4, 6 \rangle$

2

Magnitude: $||\vec{v}|| = \sqrt{4^2 + 6^2} = \sqrt{16+36} = \sqrt{52} = 2\sqrt{13}$

3

Direction: $\theta_v = \tan^{-1}(\frac{6}{4}) \approx 56.31^\circ$

2. Visual Vector Addition

To add vectors visually, use the Tip-to-Tail Method. Move the second vector so its start ("tail") touches the end ("tip") of the first vector.

The Workflow for $3A + 2B$:

  1. Sketch Vector $A$ starting from a point. Repeat it 3 times in a row.
  2. At the very end of the 3rd $A$, start Vector $B$. Repeat it 2 times.
  3. Draw a final arrow from the very beginning to the very end. That is your Resultant Vector!

Algebraic Addition

$\vec{u} = \langle 3, 4 \rangle, \vec{v} = \langle -2, 0 \rangle$

$\vec{u} + \vec{v} = \langle 3 + (-2), 4 + 0 \rangle = \langle 1, 4 \rangle$

Scalar Multiplication

$2\vec{u} - 3\vec{v} = 2\langle 3, 4 \rangle - 3\langle -2, 0 \rangle$

$= \langle 6, 8 \rangle - \langle -6, 0 \rangle = \langle 12, 8 \rangle$

Final Practice (Page 2 Guide)

Given: $\vec{u} = \langle 3, 4 \rangle$ and $\vec{v} = \langle -2, 0 \rangle$

Exercise A: $\vec{u} + \vec{v}$

  • Resultant: $\langle 1, 4 \rangle$
  • Magnitude: $||\vec{r}|| = \sqrt{1^2 + 4^2} = \sqrt{17}$
  • Direction: $\theta \approx \tan^{-1}(4/1) \approx 75.96^\circ$

Exercise B: $2\vec{u} - 3\vec{v}$

  • Resultant: $\langle 12, 8 \rangle$
  • Magnitude: $||\vec{r}|| = \sqrt{12^2 + 8^2} = \sqrt{208} = 4\sqrt{13}$
  • Direction: $\theta \approx \tan^{-1}(8/12) \approx 33.69^\circ$
Math 130 Worksheet Resource • Created for Interactive Learning