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Chapter 3: Problem 55

The sum of two vectors, \(\vec{A}+\vec{B},\) is perpendicular to their difference, \(\vec{A}-\vec{B}\). How do the vectors' magnitudes compare?

Short Answer

Expert verified
The magnitudes of the two vectors are equal.

Step by step solution

01

Write out the dot product equation

Start with the dot product equation of the sum and the difference of the vectors \(\vec{A}\) and \(\vec{B}\), which should be equal to zero since they are perpendicular: \((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = 0\)
02

Expand the dot product

Expand this dot product. Remember the distributive aspect of the dot product. This results in: \(\vec{A} \cdot \vec{A} - \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} - \vec{B} \cdot \vec{B} = 0\)
03

Simplify the equation

Notice that \(\vec{A} \cdot \vec{B}\) is the same as \(\vec{B} \cdot \vec{A}\). Simplify the equation accordingly to get: \(||\vec{A}||^2 - ||\vec{B}||^2 = 0\)
04

Interpret the result

From the last equation, we see that the square of the magnitude of vector \(\vec{A}\) has to be equal to the square of the magnitude of vector \(\vec{B}\) for their sum to be perpendicular to their difference.

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