Problem 83 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: Write down the vectors

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

Step 2: Calculate the dot product

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

Step 3: 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 \(b = -60 / -3 = 20\).