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Chapter 6: 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 orthogonal is \(b = -20\).

Step by step solution

01

Write down the two given vectors

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

Set up the dot product equation

The dot product of two vectors \(\mathbf{A} = A_{x}\mathbf{i} + A_{y}\mathbf{j}\) and \(\mathbf{B} = B_{x}\mathbf{i} + B_{y}\mathbf{j}\) is given by \(A_{x}B_{x} + A_{y}B_{y} = 0\). Substitute \(A_{x} = 15, A_{y} = -3, B_{x} = -4, B_{y} = b\) into the equation and set it equal to zero because the vectors are orthogonal.
03

Solve for b

The equation from step 2 becomes \(15*(-4) - 3*b = 0\). Solving this equation leads to \(b = -20\).

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