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Chapter 6: Problem 27

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=2 \mathbf{i}-2 \mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}$$

Short Answer

Expert verified
The vectors \(\mathbf{v}\) and \(\mathbf{w}\) are not orthogonal because their dot product is not zero.

Step by step solution

01

Identify the components of the vectors

The vector \(\mathbf{v}\) has i-component equal to 2 and j-component equal to -2. The vector \(\mathbf{w}\) has i-component equal to -1 and j-component equal to 1.
02

Compute the dot product

The dot product of the vectors \(\mathbf{v}\) and \(\mathbf{w}\) is calculated by multiplying corresponding i-components and j-components and summing them. \(\mathbf{v} \cdot \mathbf{w} = (2 * -1) + (-2 * 1) = -2-2 = -4\).
03

Check if vectors are orthogonal

The vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal if and only if their dot product is equal to zero. Since the dot product \(\mathbf{v} \cdot \mathbf{w} = -4\), which is not equal to zero, we conclude that \(\mathbf{v}\) and \(\mathbf{w}\) are not orthogonal.

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