https://file.aiquickdraw.com/mathbot/file/1734264968073n87zthjg.jpg

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

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### 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:

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### 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:
\[
\overrightarrow{\mathbf{CD}} = 0 \mathbf{p}
\]

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### 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:
\[
\overrightarrow{\mathbf{DC}} = 0 \mathbf{p}
\]

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### 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 \( \mathbf{p} \).

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### 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.

Express the following in terms of p: a) CD (vector) b) DC (vector). Give your answers in their simplest forms.

Learn how to express vectors CD and DC in terms of a given vector p. This problem focuses on vector components, perpendicular directions, and vector relationships. Suitable for students in Grades 8-10 studying vector mathematics.

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