Vector Diagrams

What is a Vector Diagram?

A vector diagram shows both the magnitude (length) and direction of a vector.

• Vectors are drawn as arrows.
• The length of the arrow represents the magnitude.
• The arrowhead shows the direction.

If a vector goes from A to B, we write it as: $\vec{AB}$

This points from A toward B.

Drawing Vectors on a Grid

You can draw a vector anywhere on a coordinate grid, as long as it has the same movement.

Example:

$$\vec{q} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}$$

• Start at any point.
• Move 2 left and 5 up.
• Draw an arrow in that direction.

Example with zero:

$$\vec{r} = \begin{pmatrix} 0 \\ -3 \end{pmatrix} \quad \text{means no movement on the right 3 down}$$

Multiplying Vectors by Scalars

Multiplying a vector by a number (called a scalar) changes its size:

• Positive scalar: same direction, longer or shorter arrow
• Negative scalar: reverse direction

Example:

$$\vec{v} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$$

$$2\vec{v} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}, \quad -\vec{v} = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$$

Adding and Subtracting Vectors (Graphically)

To add vectors $\vec{a} + \vec{b}$:

• Draw $\vec{a}$
• Then draw $\vec{b}$ starting at the end of $\vec{a}$
• The total vector goes from the start of $\vec{a}$ to the end of $\vec{b}$

To subtract $\vec{a} - \vec{b}$:

• Draw $\vec{a}$
• Then draw $-\vec{b}$, which is $\vec{b}$ in the opposite direction
• Draw the vector from the start of $\vec{a}$ to the end of $-\vec{b}$

 
Example

Points A, B, and C are on a grid.

(a) Write vectors $\vec{AB}$, $\vec{AC}$, and $\vec{CB}$ as column vectors.

$\vec{AB} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$

$\vec{AC} = \begin{pmatrix} 7 \\ -6 \end{pmatrix}$

$\vec{CB} = \begin{pmatrix} -1 \\ 8 \end{pmatrix}$

(b) Explain why:

$$\vec{AB} + \vec{BC} + \vec{CA} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

This is a loop: $A \to B \to C \to A$

Since it returns to the starting point, the overall movement is zero.