Finding Vector Paths

 

Understanding Vector Paths

A vector path is a collection of vectors that will take you from a start point to an end point.

In some vector problems, you're given a grid made up of parallelograms and asked to describe a path from one point to another using given vectors like $\mathbf{a}$ and $\mathbf{b}$.

These paths can be written by combining multiples of $\mathbf{a}$, $\mathbf{b}$, or their opposites:

• $\mathbf{a}$: usually represents one step right
• $-\mathbf{a}$: one step left
• $\mathbf{b}$: one diagonal step up and right
• $-\mathbf{b}$: one diagonal step down and left

Each move counts as a vector step, and a full journey can be described as:

$$\text{Path Vector} = m\mathbf{a} + n\mathbf{b}$$

Where:

• $m$ is the number of horizontal steps
• $n$ is the number of diagonal steps

 

Examples

1. From A to E

If A to E involves 4 right steps, then:

$$\overrightarrow{AE} = 4\mathbf{a}$$

 

2. From G to T

One possible route:

2 diagonal up-right steps $\to 2\mathbf{b}$

3 right steps $\to 3\mathbf{a}$

So,

$$\overrightarrow{GT} = 3\mathbf{a} + 2\mathbf{b}$$

 

3. From E to K

Try this route:

2 diagonal up-right steps \(\to \  2\mathbf{b}\)

4 left steps \(\to \  -4\mathbf{a}\)

So,

$$\overrightarrow{EK} = -4\mathbf{a} + 2\mathbf{b}$$