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Chapter 6: Problem 31

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

Short Answer

Expert verified
Yes, the vectors v and w are orthogonal, since their dot product equals to zero.

Step by step solution

01

Identify the components of the vectors

From the equation, it can be seen that the vector \( \mathbf{v} \) has components (3, 0) and vector \( \mathbf{w} \) has components (0, -4).
02

Compute the dot product

The dot product of two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \). Here, \( \mathbf{v} \cdot \mathbf{w} = (3)(0) + (0)(-4) = 0 \).
03

Determining Orthogonality

Since the dot product calculated equals to zero, it can be concluded that vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal.

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Most popular questions from this chapter

Verify the identity: $$\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4 \tan x \sec x$$ (Section 5.1, Example 5)

Draw two vectors, \(\mathbf{v}\) and \(\mathbf{w},\) with the same initial point. Show the vector projection of \(\mathbf{v}\) onto \(\mathbf{w}\) in your diagram. Then describe how you identified this vector.

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

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

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}+\mathbf{w})$$
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