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Chapter 7: Problem 83

Find a value of \(b\) so that \(15 \mathbf{i}-3 \mathbf{j}\) and \(-4 \mathbf{i}+b \mathbf{j}\) are orthogonal.

Short Answer

Expert verified
The value of \(b\) that makes the vectors \(15 \mathbf{i}-3 \mathbf{j}\) and \(-4 \mathbf{i}+b \mathbf{j}\) orthogonal is \(b = 20\).

Step by step solution

01

Write down the vectors

The given vectors are \(15 \mathbf{i}-3 \mathbf{j}\) and \(-4 \mathbf{i}+b \mathbf{j}\)
02

Calculate the dot product

The dot product of the two vectors is given by \(15*-4 + (-3*b) = -60 -3b\)
03

Set the dot product equal to zero and solve for \(b\)

To find the values of \(b\) that make the vectors orthogonal, set the dot product equal to zero and solve for \(b\). This gives the equation -60 -3b = 0. Solving this for \(b\) yields \(b = -60 / -3 = 20\).

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