Question

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors. The angle between \( \vec{a} \) and \( \vec{b} \) is \( 30^\circ \), the angle between \( \vec{a} \) and \( \vec{b} + \vec{c} \) is \( 45^\circ \). If \( |\vec{b}| = \sqrt{6} \) and \( |\vec{c}| = 2\sqrt{2} \), then \( |\vec{b} + \vec{c}| \) is:

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

In problems involving the magnitudes of vector sums, use the Law of Cosines and dot product relations to connect the given angles and magnitudes. This helps in solving for unknowns effectively.

Answer 1

Answer: (E)

We are given the following information:
- The angle between \( \vec{a} \) and \( \vec{b} \) is \( 30^\circ \),
- The angle between \( \vec{a} \) and \( \vec{b} + \vec{c} \) is \( 45^\circ \), - \( |\vec{b}| = \sqrt{6} \), - \( |\vec{c}| = 2\sqrt{2} \). We need to find \( |\vec{b} + \vec{c}| \).
Step 1: Use the Law of Cosines
First, use the Law of Cosines to express \( |\vec{b} + \vec{c}| \). The formula for the magnitude of the sum of two vectors is: \[ |\vec{b} + \vec{c}|^2 = |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{b}||\vec{c}|\cos(\theta) \] where \( \theta \) is the angle between \( \vec{b} \) and \( \vec{c} \).
We are not directly given the angle between \( \vec{b} \) and \( \vec{c} \), but we can use the information about the angle between \( \vec{a} \) and \( \vec{b} + \vec{c} \).
Step 2: Use the angle between \( \vec{a} \) and \( \vec{b} + \vec{c} \)
The angle between \( \vec{a} \) and \( \vec{b} + \vec{c} \) is given as \( 45^\circ \). The dot product formula can be used:
\[ \vec{a} \cdot (\vec{b} + \vec{c}) = |\vec{a}| |\vec{b} + \vec{c}| \cos(45^\circ) \] This equation allows us to find the relationship between the magnitudes of \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \). After solving this system of equations, we find that: \[ |\vec{b} + \vec{c}| = 5 \] Thus, the correct answer is option (E), \( |\vec{b} + \vec{c}| = 5 \).