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Chapter 6: Problem 23

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
Yes, the vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal as their dot product is zero.

Step by step solution

01

Writing the Vector Form

Vectors \(\mathbf{v}\) and \(\mathbf{w}\) can be written as \[\mathbf{v} = 1\mathbf{i}+1\mathbf{j}\], \[\mathbf{w} = 1\mathbf{i}-1\mathbf{j}\] respectively
02

Computing the Dot Product

The dot product of two vectors is given by \[\mathbf{v} . \mathbf{w} = (v_1 * w_1) + (v_2 * w_2)\] where \(v_1\), \(v_2\) are the coefficients of vector \(\mathbf{v}\) in \(i\), \(j\) direction and \(w_1\), \(w_2\) are coefficients of vector \(\mathbf{w}\) in \(i\), \(j\) direction respectively. So in this case, \[\mathbf{v} . \mathbf{w} = (1 * 1) + (1 * -1) = 0\]
03

Concluding the Orthogonality

Since the dot product of vectors \(\mathbf{v}\) and \(\mathbf{w}\) is zero, these vectors are orthogonal to each other.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Under certain conditions, a fire can be located by superimposing a triangle onto the situation and applying the Law of sines.

Find two vectors \(\mathbf{v}\) and \(\mathbf{w}\) such that the projection of \(\mathbf{v}\) onto \(\mathbf{w}\) is \(\mathbf{v}\).

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 \(\mathbf{w}.\) $$\mathbf{v}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+3 \mathbf{j}$$

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a polar equation that failed the symmetry test with respect to \(\theta=\frac{\pi}{2},\) so my graph will not have this kind of symmetry.
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