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Chapter 12: Problem 5

Explain how to find the angle between two nonzero vectors.

Short Answer

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Question: Explain the process of finding the angle between two nonzero vectors. Answer: To find the angle between two nonzero vectors, follow these steps: 1. Understand the dot product formula, which involves the cosine of the angle between the vectors: \(A \cdot B = \|A\| \|B\| \cos{\theta}\) 2. Calculate the dot product by multiplying the corresponding components of the two vectors and adding the products: \(A \cdot B = a_1 b_1 + a_2 b_2 + a_3 b_3\). 3. Calculate the magnitudes of the vectors: \(\|A\| = \sqrt{a_1^2 + a_2^鈧� + a_3^2}\) and \(\|B\| = \sqrt{b_1^2 + b_2^鈧� + b_3^2}\). 4. Find the cosine of the angle by dividing the dot product by the product of the magnitudes: \(\cos{\theta} = \frac{A \cdot B}{\|A\| \|B\|}\). 5. Calculate the angle using the inverse cosine (arccos) function: \(\theta = \arccos{\left(\frac{A \cdot B}{\|A\| \|B\|}\right)}\).

Step by step solution

01

Understanding the Dot Product

Let two nonzero vectors be A and B. The dot product of the vectors A and B is defined as: \(A \cdot B = \|A\| \|B\| \cos{\theta}\), where \(\|A\|\) and \(\|B\|\) are the magnitudes of the vectors A and B, respectively, and \(\theta\) is the angle between them. Our goal is to find the angle \(\theta\).
02

Calculating the Dot Product

To calculate the dot product, multiply the corresponding components of the two vectors and then add the products. For example, if \(A = (a_1, a_2, a_3)\) and \(B = (b_1, b_2, b_3)\), the dot product is: \(A \cdot B = a_1 b_1 + a_2 b_2 + a_3 b_3\).
03

Calculating Magnitudes

Next, compute the magnitudes of vectors A and B. The magnitude of a vector is the square root of the sum of the squares of its components. If \(A = (a_1, a_2, a_3)\) and \(B = (b_1, b_2, b_3)\), then the magnitudes are: \(\|A\| = \sqrt{a_1^2 + a_2^鈧� + a_3^2}\) \(\|B\| = \sqrt{b_1^2 + b_2^鈧� + b_3^2}\)
04

Finding the Cosine of the Angle

Using the dot product formula \(A \cdot B = \|A\| \|B\| \cos{\theta}\), we can find the cosine of the angle \(\theta\) between the vectors. Divide both sides of the equation by \(\|A\| \|B\|\): \(\cos{\theta} = \frac{A \cdot B}{\|A\| \|B\|}\)
05

Finding the Angle

Now that we have the cosine of the angle, we can find the angle itself using the inverse cosine (also known as arccos) function: \(\theta = \arccos{\left(\frac{A \cdot B}{\|A\| \|B\|}\right)}\) This gives us the angle between the vectors A and B. Using these steps, you can find the angle between any two nonzero vectors.

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