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Chapter 6: Problem 24

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

Short Answer

Expert verified
The vectors \(\mathbf{v}=(1, 1)\) and \(\mathbf{w}=(-1, 1)\) are orthogonal to each other.

Step by step solution

01

Understand The Components of Each Vector

\(\mathbf{v}\) and \(\mathbf{w}\) are given as follows: \(\mathbf{v}=\mathbf{i}+\mathbf{j}\), \(\mathbf{w}=-\mathbf{i}+\mathbf{j}\). This means that \(\mathbf{v} = (1, 1)\) and \(\mathbf{w} = (-1, 1)\) in the Cartesian coordinate form.
02

Compute The Dot Product

The dot product of two vectors \(\mathbf{v} \cdot \mathbf{w}\) is calculated as \(v_1*w_1 + v_2*w_2\). Substituting the values from step 1, we have \(\mathbf{v} \cdot \mathbf{w} = (1)(-1) + (1)(1) = -1 + 1 = 0\).
03

Determine Orthogonality

Two vectors are orthogonal to each other if their dot product is zero. Since we found from step 2 that \(\mathbf{v} \cdot \mathbf{w} = 0\), we can conclude that the given vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal to each other.

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Most popular questions from this chapter

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+\frac{18}{5} \mathbf{j}$$

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