21. Given two vectors \(\mathbf{V} = \langle 20, 50 \rangle\) and \(\mathbf{W} = \langle 12, 15 \rangle\), which of the following vectors is perpendicular to \(\mathbf{W}\)?

A. \(\text{Proj}_{\mathbf{V}} \) (vector projection of \(\mathbf{V}\) onto \(\mathbf{W}\))
B. \(\text{Proj}_{\mathbf{W}} \) (vector projection of \(\mathbf{W}\) onto \(\mathbf{V}\))
C. \(\mathbf{V} - \text{Proj}_{\mathbf{V}} \)
D. \(\mathbf{V} - \text{Proj}_{\mathbf{W}} \)
E. Not listed

The vectors that are perpendicular to W are V - Proj. V and V - Proj.W. These are the orthogonal projections of V onto the line perpendicular to W. To find a vector that's perpendicular to a given vector, subtract the projection of your original vector onto the given vector from your original vector.

Explanation:

To find a vector that is perpendicular to W, we need a vector that satisfies the condition of dot product being equal to zero. This is because the dot product of two vectors is zero if and only if the vectors are orthogonal (perpendicular).

The vector projection of V onto W, denoted by Proj. V, and the vector projection of W onto V, denoted by Proj. W, are not perpendicular to W because their dot product with W is not zero.

The vectors V - Proj. V and V - Proj.W are the orthogonal projections of V onto the line perpendicular to W, and therefore are perpendicular to W. So, the correct answer is options (iii) and (iv).

In general, to find a vector that is perpendicular to a given vector, you need to find a vector that is orthogonal to the given one. This is achieved by subtracting the projection of your original vector onto the given vector from your original vector.

Learn more about Vector Perpendicularity here:

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