Math Problem Statement
Solution
To express the vectors CD and DC in terms of the given vector , let’s analyze the problem step by step.
Analyzing the Diagram:
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Vectors :
- From the image, vector is given as going from to horizontally to the left.
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Direction of Other Vectors:
- C and D are shown on the grid.
- C is positioned further down from point , creating the vector .
Now let’s determine the requested vectors:
a) Expressing :
The vector represents the displacement from to .
Since is above , this vector points vertically upward. Observing the grid:
- The vector is vertical, while is horizontal.
Thus, is perpendicular to , and there’s no component of in this direction. Therefore:
b) Expressing :
The vector is the reverse of , meaning it points vertically downward.
Since , the reverse vector is:
Final Answers:
- a)
- b)
This is because the vectors are vertical and have no components in the direction of .
Follow-up Questions:
- How do we decompose a vector into horizontal and vertical components?
- What is the significance of perpendicular vectors in terms of components?
- How do you represent vectors using unit vectors?
- How do you find the magnitude of a diagonal vector on a grid?
- Can you express 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