Projection Problems

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Objectives:

1.: Know what the projection looks like geometrically and how to compute it.

Projections: Idea and Formula

Watch the two videos “Projections: Idea and Geometry” and “Derivation of Projection Formula” in the previous module. If you need more clarification, view the two worked examples in the previous module as well.

Test your understanding with the following question.

If $\textbf {v}$ and $\textbf {w}$ are two nonzero vectors, then which of the following is true of the projection $\text {proj}_{\textbf {w}} \textbf {v}$ ? Select all that apply.

If it is nonzero, then it is always parallel to $\textbf {w}$ . $\text {proj}_{\textbf {w}} \textbf {v} = \frac {\textbf {v} \cdot \textbf {w}}{\textbf {w} \cdot \textbf {w}} \textbf {w}$ . If it is nonzero, then it is always parallel to $\textbf {v}$ .

Problems

Recall the formulas:

If $\textbf {v}$ and $\textbf {w}$ are nonzero vectors, then:

(a): The component $\displaystyle \text {comp}_{\textbf {w}} \textbf {v} = \frac {\textbf {v} \cdot \textbf {w}}{\| \textbf {w} \|}$ , and this is $\pm$ the length of the projection of $\textbf {v}$ onto $\textbf {w}$ . The book refers to this as a scalar projection.
(b): The (vector) projection $\displaystyle \text {proj}_{\textbf {w}} \textbf {v} = \frac {\textbf {v} \cdot \textbf {w}}{\textbf {w} \cdot \textbf {w}} \textbf {w}$ .

Let $\textbf {v} = \left \langle 1,2,0 \right \rangle$ an $\textbf {w} = \left \langle 3,5,1 \right \rangle$ .

The dot product of $\textbf {v}$ and $\textbf {w}$ is $\textbf {v} \cdot \textbf {w} = \answer {13}$ .
The angle between $\textbf {v}$ and $\textbf {w}$ is acuterightobtuse .
The projection $\text {proj}_{\textbf {w}} \textbf {v}$ will be in the direction as $\textbf {w}$ .
The component (i.e. scalar projection) $\text {comp}_{\textbf {w}} \textbf {v} = \answer {\frac {13}{\sqrt {35}}}$ .
The projection $\text {proj}_{\textbf {w}} \textbf {v} = \left \langle \answer {\frac {39}{35}}, \answer {\frac {65}{35}},\answer {\frac {13}{35}} \right \rangle .$

Suppose $\textbf {v}$ and $\textbf {w}$ are nonzero three-dimensional vectors with $\textbf {v}$ orthogonal to $\textbf {w}$ . Then $\text {proj}_{\textbf {w}} \textbf {v} = \left \langle \answer {0},\answer {0},\answer {0} \right \rangle .$

If $\textbf {v}$ is orthogonal to $\textbf {w}$ , then what is $\textbf {v} \cdot \textbf {w}$ ? Use this in the projection formula.

True/False

If $\textbf {v}$ and $\textbf {w}$ are nonzero vectors and $\textbf {v} = \textbf {w}$ , then it must be true that $\text {proj}_{\textbf {w}} \textbf {v} = \textbf {v}.$ TrueFalse

Assume $\textbf {v}$ and $\textbf {w}$ are nonzero vectors. Then it must be true that $\text {proj}_{2\textbf {w}} \textbf {v} = 2\text {proj}_{\textbf {w}}\textbf {v}.$ TrueFalse