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Chapter 11: Problem 5

Explain how to find the angle between two nonzero vectors.

Short Answer

Expert verified
Answer: To find the angle between two nonzero vectors, follow these steps: 1. Write the given vectors A and B in component form (A_x, A_y, A_z) and (B_x, B_y, B_z). 2. Calculate the dot product A 鈰� B. 3. Calculate the magnitude ||A|| and ||B||. 4. Use the formula for the angle between the vectors: \( cos(\theta) = \frac{A \cdot B}{||A|| ||B||} \). 5. Find the inverse cosine (arccos) of the obtained cosine value to get the angle 胃. 6. Express the angle in degrees or radians, as required.

Step by step solution

01

Understand the concept of vector dot product

The dot product (also known as the scalar or inner product) is a fundamental operation in vector algebra. Given two vectors, it is defined as the sum of the product of their corresponding components. For two vectors A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z) in three-dimensional coordinate space, the dot product is given as: \[ A \cdot B = A_xB_x + A_yB_y + A_zB_z \]
02

Understand the concept of magnitude of a vector

The magnitude of a vector, also known as the length or norm of a vector, is a measure of the vector's size. It is found by taking the square root of the sum of the squares of the components of the vector. For a vector A = (A_x, A_y, A_z) in three-dimensional coordinate space, the magnitude of A is: \[ ||A|| = \sqrt{A_x^2 + A_y^2 + A_z^2} \]
03

Calculate the angle between two vectors

The angle 胃 between two nonzero vectors A and B can be found using the relationship between the dot product and the magnitudes of the vectors. The formula for the angle 胃 is given as: \[ cos(\theta) = \frac{A \cdot B}{||A|| ||B||} \] Where 胃 is the angle between vectors A and B.
04

Follow the steps to find the angle between two nonzero vectors

In order to find the angle between two nonzero vectors, follow these steps: 1. Write the given vectors A and B in component form (A_x, A_y, A_z) and (B_x, B_y, B_z). 2. Calculate the dot product A 鈰� B. 3. Calculate the magnitude ||A|| and ||B||. 4. Use the formula for the angle between the vectors: \( cos(\theta) = \frac{A \cdot B}{||A|| ||B||} \). 5. Find the inverse cosine (arccos) of the obtained cosine value to get the angle 胃. 6. Express the angle in degrees or radians, as required. Following these steps, you can find the angle between two nonzero vectors.

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Most popular questions from this chapter

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