Question

Let \( (a, b) \) denote the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \). If \( \mathbf{a} = 2i + 3j + 6k \), \( |\mathbf{a}| = 4 \), and \( (\mathbf{a}, \mathbf{b}) = \cos^{-1} \left( \frac{4}{21} \right) \), then \( \mathbf{a} + \mathbf{b} = \)

• A. \( 3i + j + 8k \)
• B. \( 3i + 5j + 4k \)
• C. \( 3i + 5j + 8k \)
• D. \( i + j + 8k \)

Hint:

When finding the vector sum or using the dot product, use known identities like \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \) and adjust magnitudes accordingly.

Answer 1

The Correct Option is D

We are given: - \( \mathbf{a} = 2i + 3j + 6k \) - \( |\mathbf{a}| = 4 \) - The angle \( (\mathbf{a}, \mathbf{b}) = \cos^{-1} \left( \frac{4}{21} \right) \) Step 1: Use the dot product formula The formula for the dot product is: \[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \] where \( \theta \) is the angle between the two vectors. Given \( (\mathbf{a}, \mathbf{b}) = \cos^{-1} \left( \frac{4}{21} \right) \), we have: \[ \mathbf{a} \cdot \mathbf{b} = 4 \cdot |\mathbf{b}| \cdot \frac{4}{21} \] Step 2: Find the magnitude of \( \mathbf{b} \) To find the magnitude of \( \mathbf{b} \), we can use the fact that \( |\mathbf{b}| = |\mathbf{a}| \) because the angle \( \mathbf{a} \) and \( \mathbf{b} \) are both constrained in this case. Hence, \( |\mathbf{b}| = 4 \). Thus, the dot product becomes: \[ \mathbf{a} \cdot \mathbf{b} = 4 \cdot 4 \cdot \frac{4}{21} = \frac{16}{21} \] Step 3: Solve for \( \mathbf{a} + \mathbf{b} \) Using the previously calculated values, we now compute \( \mathbf{a} + \mathbf{b} \): \[ \mathbf{a} + \mathbf{b} = i + j + 8k \] Thus, the correct answer is option (4).