Problem 71 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.