Question

If \(\bar{a}, \bar{b}, \bar{c}\) are three unit vectors such that \(|\bar{a} + \bar{b}|^2 + |\bar{a} + \bar{c}|^2 = 8\), then \(|\bar{a} + 3\bar{b}|^2 + |\bar{a} + 3\bar{c}|^2 =\)

• A. \(26\)
• B. \(32\)
• C. \(22\)
• D. \(36\)

Hint:

Whenever squared magnitudes of vector sums are given, convert them into dot products. It makes the calculation much easier.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Unit vectors have magnitude \(1\). We use the property of the dot product where \(|\bar{u} + \bar{v}|^2 = |\bar{u}|^2 + |\bar{v}|^2 + 2\bar{u} \cdot \bar{v}\).
Step 2: Key Formula or Approach:
For unit vectors \(\bar{a}, \bar{b}, \bar{c}\), \(|\bar{a}|^2 = |\bar{b}|^2 = |\bar{c}|^2 = 1\).
Equation: \(|\bar{a} + \bar{b}|^2 = 1 + 1 + 2\bar{a} \cdot \bar{b} = 2 + 2\bar{a} \cdot \bar{b}\).
Step 3: Detailed Explanation:
Given: \((2 + 2\bar{a} \cdot \bar{b}) + (2 + 2\bar{a} \cdot \bar{c}) = 8\)
\[ 4 + 2(\bar{a} \cdot \bar{b} + \bar{a} \cdot \bar{c}) = 8 \implies 2(\bar{a} \cdot \bar{b} + \bar{a} \cdot \bar{c}) = 4 \implies \bar{a} \cdot \bar{b} + \bar{a} \cdot \bar{c} = 2 \] Now calculate the required expression:
\[ |\bar{a} + 3\bar{b}|^2 + |\bar{a} + 3\bar{c}|^2 \] \[ = (|\bar{a}|^2 + 9|\bar{b}|^2 + 6\bar{a} \cdot \bar{b}) + (|\bar{a}|^2 + 9|\bar{c}|^2 + 6\bar{a} \cdot \bar{c}) \] \[ = (1 + 9 + 6\bar{a} \cdot \bar{b}) + (1 + 9 + 6\bar{a} \cdot \bar{c}) \] \[ = 20 + 6(\bar{a} \cdot \bar{b} + \bar{a} \cdot \bar{c}) \] Substituting \(\bar{a} \cdot \bar{b} + \bar{a} \cdot \bar{c} = 2\):
\[ = 20 + 6(2) = 20 + 12 = 32 \] Step 4: Final Answer:
The result is \(32\).