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Chapter 6: Problem 32

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=-6 \mathbf{j}$$

Short Answer

Expert verified
Yes, the vectors \(v\) and \(w\) are orthogonal because their dot product is zero.

Step by step solution

01

Define the vectors

The given vectors are \(\mathbf{v}=5 \mathbf{i}\) and \(\mathbf{w}=-6 \mathbf{j}\). The task is to calculate the dot product between these two vectors.
02

Calculate the dot product

The dot product between two vectors \(\mathbf{A}=A_x \mathbf{i}+A_y \mathbf{j}\) and \(\mathbf{B}=B_x \mathbf{i}+B_y \mathbf{j}\) is defined as \( \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y \). In our case, we have \(\mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y = 5 * 0 + 0 * -6 = 0.\)
03

Determine if vectors are orthogonal

The vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal if their dot product is 0. Since we found that \(\mathbf{v} \cdot \mathbf{w} = 0\), the vectors are orthogonal.

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Most popular questions from this chapter

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+3 \mathbf{j}$$

A force is given by the vector \(\mathbf{F}=3 \mathbf{i}+2 \mathbf{j} .\) The force moves an object along a straight line from the point (4,9) to the point \((10,20) .\) Find the work done if the distance is measured in feet and the force is measured in pounds.

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=3 \mathbf{i}, \quad \mathbf{w}=-4 \mathbf{i}$$

What are orthogonal vectors?

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=2 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}$$
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