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Chapter 6: Problem 28

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}$$

Short Answer

Expert verified
No, the vectors v and w are not orthogonal as their dot product does not equal zero.

Step by step solution

01

Unpack the vectors

The vectors v and w are given in the problem. We need to write them out in their component form. Vector v is \(5\mathbf{i}- 5\mathbf{j}\) so the components of v are 5 and -5. Similarly, the components of vector w, which is \(\mathbf{i} - \mathbf{j}\), are 1 and -1.
02

Compute the dot product

The dot product is found by multiplying corresponding components in the two vectors, and summing those products. The dot product of vectors v and w is given by \(5*1 + (-5)*(-1) = 5 + 5 =10\).
03

Assess Orthogonality

Two vectors are orthogonal if and only if their dot product equals 0. As the dot product we calculated is 10 and not 0, we thus conclude that the vectors v and w are not orthogonal.

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