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Chapter 6: Problem 25

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=2 \mathbf{i}+8 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j}$$

Short Answer

Expert verified
Yes, \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal as their dot product is 0.

Step by step solution

01

Determining the components of vectors

Each vector is represented in a two-dimensional space, and we can represent these vectors in component form. For vector \(\mathbf{v}\), the components are \(2\) and \(8\) and for vector \(\mathbf{w}\), the components are \(4\) and \(-1\)
02

Calculate the dot product of \(\mathbf{v}\) and \(\mathbf{w}\)

The dot product of two vectors \(\mathbf{v}\) and \(\mathbf{w}\) can be calculated by multiplying corresponding components of two vectors, and then summing up these products. The dot product of \(\mathbf{v} = 2\mathbf{i} + 8\mathbf{j}\) and \(\mathbf{w} = 4\mathbf{i} - \mathbf{j}\) can be computed as follows: \( (2)(4) + (8)(-1) = 8 - 8 = 0 \)
03

Interpret the result

Since the dot product of \(\mathbf{v}\) and \(\mathbf{w}\) equals to 0, these vectors are orthogonal

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