Math Problem Statement

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

Solution

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


Analyzing the Diagram:

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

    • From the image, vector p\mathbf{p} is given as going from AA to BB horizontally to the left.
  2. Direction of Other Vectors:

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

Now let’s determine the requested vectors:


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

The vector CD\overrightarrow{\mathbf{CD}} represents the displacement from CC to DD.

Since DD is above CC, this vector points vertically upward. Observing the grid:

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

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


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

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

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


Final Answers:

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

This is because the vectors are vertical and have no components in the direction of p\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 p\mathbf{p} in terms of another vector?

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

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Math Problem Analysis

Mathematical Concepts

Vectors
Vector Components
Direction of Vectors
Displacement

Formulas

Vector addition: \( \mathbf{AB} + \mathbf{BC} = \mathbf{AC} \)
Reversing a vector: \( \overrightarrow{DC} = -\overrightarrow{CD} \)

Theorems

Vectors that are perpendicular have no horizontal components relative to another vector.

Suitable Grade Level

Grades 8-10