To express the vectors CD and DC in terms of the given vector $\mathbf{p}$, let’s analyze the problem step by step.

Analyzing the Diagram:

1. Vectors $\mathbf{AB} = \mathbf{p}$:

   • From the image, vector $\mathbf{p}$ is given as going from $A$ to $B$ horizontally to the left.

2. Direction of Other Vectors:

   • C and D are shown on the grid.
   • C is positioned further down from point $D$, creating the vector $\overrightarrow{\mathbf{CD}}$.

Now let’s determine the requested vectors:

a) Expressing $\overrightarrow{\mathbf{CD}}$:

The vector $\overrightarrow{\mathbf{CD}}$ represents the displacement from $C$ to $D$.

Since $D$ is above $C$, this vector points vertically upward. Observing the grid:

• The vector $\overrightarrow{\mathbf{CD}}$ is vertical, while $\mathbf{p}$ is horizontal.

Thus, $\overrightarrow{\mathbf{CD}}$ is perpendicular to $\mathbf{p}$, and there’s no component of $\mathbf{p}$ in this direction. Therefore: $\overrightarrow{\mathbf{CD}} = 0 \mathbf{p}$

b) Expressing $\overrightarrow{\mathbf{DC}}$:

The vector $\overrightarrow{\mathbf{DC}}$ is the reverse of $\overrightarrow{\mathbf{CD}}$, meaning it points vertically downward.

Since $\overrightarrow{\mathbf{CD}} = 0 \mathbf{p}$, the reverse vector $\overrightarrow{\mathbf{DC}}$ is: $\overrightarrow{\mathbf{DC}} = 0 \mathbf{p}$

Final Answers:

1. a) $\overrightarrow{\mathbf{CD}} = 0 \mathbf{p}$
2. b) $\overrightarrow{\mathbf{DC}} = 0 \mathbf{p}$

This is because the vectors are vertical and have no components in the direction of $\mathbf{p}$.

Follow-up Questions:

1. How do we decompose a vector into horizontal and vertical components?
2. What is the significance of perpendicular vectors in terms of components?
3. How do you represent vectors using unit vectors?
4. How do you find the magnitude of a diagonal vector on a grid?
5. Can you express $\mathbf{p}$ in terms of another vector?

Tip: Perpendicular vectors have a dot product of zero, which is a useful property in vector calculations.