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Chapter 11: Problem 5

What's the angle between two vectors if their dot product is equal to the magnitude of their cross product?

Short Answer

Expert verified
The angle between the two vectors is \(45\) degrees.

Step by step solution

01

Determine the Dot Product

Firstly, recall the formula for the dot product of two vectors: it is equal to the product of their magnitudes and the cosine of the angle between them. Let's denote this angle by \(\theta\), and the magnitudes of the vectors as \( ||A|| \) and \( ||B|| \). So, the dot product of \( A \) and \( B \) is defined as \( A \cdot B = ||A|| \cdot ||B|| \cdot \cos(\theta) \).
02

Determine the Magnitude of the Cross Product

Next, remember the formula for the magnitude of the cross product of two vectors: it equals to the product of their magnitudes and the sine of the angle between them. Using the same notations, the magnitude of the cross product is \( ||A \times B|| = ||A|| \cdot ||B|| \cdot \sin(\theta) \).
03

Set the dot product equal to the magnitude of the cross product

According to the exercise, the dot product of the vectors is equal to the magnitude of their cross product. So, we can write the equation \( ||A|| \cdot ||B|| \cdot \cos(\theta) = ||A|| \cdot ||B|| \cdot \sin(\theta) \).
04

Simplify the equation and solve for the angle

Now, we can simplify the equation from the previous step. As the magnitudes of the vectors are non-zero, we can divide both sides of the equation by \( ||A|| \cdot ||B|| \) and get \( \cos(\theta) = \sin(\theta) \). This equation means that the angle is equal to \( \frac{\pi}{4} \) or \( 45 \) degrees, because cosine and sine are equal at these points.

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