Question

Let \( \overrightarrow{u}, \overrightarrow{v}, \overrightarrow{w} \) be three unit vectors. Let \( \overrightarrow{p} = \overrightarrow{u} + \overrightarrow{v} + \overrightarrow{w} \), \( \overrightarrow{q} = \overrightarrow{u} \times (\overrightarrow{p} \times \overrightarrow{w}) \). If \( \overrightarrow{p} \cdot \overrightarrow{u} = \frac{3}{2} \), \( \overrightarrow{p} \cdot \overrightarrow{v} = \frac{7}{4} \), \( |\overrightarrow{p}| = 2 \), and \( \overrightarrow{v} = K \overrightarrow{q} \), then \( K = ? \)

• A. \( -1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( -2 \)

Hint:

When solving vector equations, use the given dot product values and apply fundamental vector identities like the dot product and triple product expansion. This simplifies finding unknown scalar factors.

Answer 1

The Correct Option is B

We are given the unit vectors \( \overrightarrow{u}, \overrightarrow{v}, \overrightarrow{w} \) and the equations: \[ \overrightarrow{p} = \overrightarrow{u} + \overrightarrow{v} + \overrightarrow{w} \] \[ \overrightarrow{q} = \overrightarrow{u} \times (\overrightarrow{p} \times \overrightarrow{w}) \] \[ \overrightarrow{p} \cdot \overrightarrow{u} = \frac{3}{2}, \quad \overrightarrow{p} \cdot \overrightarrow{v} = \frac{7}{4}, \quad |\overrightarrow{p}| = 2 \] \[ \overrightarrow{v} = K \overrightarrow{q} \] Step 1: Analyze the Given Conditions
We know that \( \overrightarrow{p} \cdot \overrightarrow{p} = |\overrightarrow{p}|^2 \), so: \[ \overrightarrow{p} \cdot \overrightarrow{p} = 4 \] Expanding: \[ (\overrightarrow{u} + \overrightarrow{v} + \overrightarrow{w}) \cdot (\overrightarrow{u} + \overrightarrow{v} + \overrightarrow{w}) = 4 \] Expanding using dot product properties: \[ |\overrightarrow{u}|^2 + |\overrightarrow{v}|^2 + |\overrightarrow{w}|^2 + 2(\overrightarrow{u} \cdot \overrightarrow{v} + \overrightarrow{v} \cdot \overrightarrow{w} + \overrightarrow{w} \cdot \overrightarrow{u}) = 4 \] Since \( \overrightarrow{u}, \overrightarrow{v}, \overrightarrow{w} \) are unit vectors: \[ 1 + 1 + 1 + 2(\overrightarrow{u} \cdot \overrightarrow{v} + \overrightarrow{v} \cdot \overrightarrow{w} + \overrightarrow{w} \cdot \overrightarrow{u}) = 4 \] \[ 2(\overrightarrow{u} \cdot \overrightarrow{v} + \overrightarrow{v} \cdot \overrightarrow{w} + \overrightarrow{w} \cdot \overrightarrow{u}) = 1 \] Step 2: Solve for \( K \)
Using the equation \( \overrightarrow{v} = K \overrightarrow{q} \), we equate the known dot product results: \[ \overrightarrow{p} \cdot \overrightarrow{v} = K \overrightarrow{p} \cdot \overrightarrow{q} \] Substituting the given values: \[ \frac{7}{4} = K \cdot \frac{7}{4} \] Solving for \( K \): \[ K = 2 \]