https://file.aiquickdraw.com/mathbot/file/1725537449881i009d15l.jpg

The problem in the image can be interpreted as follows:

- \( A \) and \( B \) are the two components of the vector \( \mathbf{V} = \hat{i} + 2\hat{j} + 5\hat{k} \).
- The value of vector \( B \) is equal to the value of the perpendicular projection of \( \mathbf{V} \) parallel to \( \mathbf{A} = 2\hat{i} + 2\hat{j} + \hat{k} \).
- We are asked to find the angle between \( A \) and \( B \).

To approach this, we will:

1. Calculate the projection of \( \mathbf{V} \) onto \( \mathbf{A} \) using the projection formula:
   \[
   \text{Proj}_{\mathbf{A}} \mathbf{V} = \frac{\mathbf{V} \cdot \mathbf{A}}{|\mathbf{A}|^2} \mathbf{A}
   \]
2. Set the magnitude of this projection equal to the magnitude of \( B \), and then solve for the angle between \( A \) and \( B \).

Let's begin by calculating the projection and then find the angle between the vectors.Here are the results:

1. The projection of vector \( \mathbf{V} \) onto vector \( \mathbf{A} \) is approximately \( [2.44\hat{i}, 2.44\hat{j}, 1.22\hat{k}] \).
2. The magnitude of vector \( B \), which is equal to the magnitude of this projection, is approximately \( 3.67 \).
3. The angle between \( A \) and \( B \) is \( 0^\circ \), meaning they are in the same direction.

This indicates that \( A \) and \( B \) are collinear.

Would you like further clarification or details? Here are 5 follow-up questions you might explore:

1. How would the angle change if the vectors were not collinear?
2. Can you find the cross product of \( A \) and \( B \) to confirm collinearity?
3. What happens if \( \mathbf{V} \) had a different form or components?
4. How would you interpret the perpendicular projection in a different scenario?
5. How can the dot product be used to understand the relationship between two vectors?

**Tip:** The dot product of two vectors is a key tool in determining their angle. When the dot product is zero, the vectors are orthogonal (perpendicular).

A and B are the two components of vector V = i + 2j + 5k. If the value of the vector B is equal to the value of the perpendicular projection of V parallel to A = 2i + 2j + k, find the angle between A and B.

Solve a vector problem involving the projection of vector V = i + 2j + 5k onto A = 2i + 2j + k to find the angle between vectors A and B. This problem covers vector projection, magnitude calculation, and angle determination using the dot product.

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