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Chapter 7: Problem 71

Draw two vectors, \(\mathbf{v}\) and \(\mathbf{w},\) with the same initial point. Show the vector projection of \(\mathbf{v}\) onto \(\mathbf{w}\) in your diagram. Then describe how you identified this vector.

Short Answer

Expert verified
The vector projection of \(\mathbf{v}\) onto \(\mathbf{w}\) is represented diagonally on the diagram towards \(\mathbf{w}\), which is calculated based on the dot product of \(\mathbf{v}, \mathbf{w}\) divided by the magnitude square of \(\mathbf{w}\) then multiplied by the \(\mathbf{w}\) vector. This vector lies on the line defined by \(\mathbf{w}\) with the same initial point.

Step by step solution

01

Drawing two vectors

First, draw two vectors (\(\mathbf{v}\) and \(\mathbf{w}\)) on a cartesian coordinate system with the same initial point. You can choose any convenient points in the system to define these vectors.
02

Showing vector projection

Now, the vector projection of \(\mathbf{v}\) onto \(\mathbf{w}\) (written as proj_\(\mathbf{w}\)(\(\mathbf{v}\))) is given by (\(\mathbf{v} . \mathbf{w}\))/\(\|\mathbf{w}\|^2\) * \(\mathbf{w}\), where '\(.\)' denotes the dot product and '\(\|\mathbf{w}\|\)' is the magnitude of vector \(\mathbf{w}\). After calculating, draw this resultant vector in the diagram, which should be on the line of vector \(\mathbf{w}\) starting from the same initial point.
03

Illustrate the identified vectors

Label the vectors \(\mathbf{v}, \mathbf{w}\), and the projection vector in the diagram. Moreover, the projection vector would lie entirely along vector \(\mathbf{w}\) and possibly shorter.

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