Question

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \), \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \), \( \vec{c} = \lambda \hat{i} + \hat{j} + \hat{k} \) and \( \vec{v} = \vec{a} \times \vec{b} \). If \( \vec{v} \cdot \vec{c} = 11 \) and the length of the projection of \( \vec{b} \) on \( \vec{c} \) is \( p \), then find the value of \( 9p^2 \).

• A. 12
• B. 4
• C. 9
• D. 6

Hint:

Scalar Triple Product $[\vec{a} \vec{b} \vec{c}] = (\vec{a} \times \vec{b}) \cdot \vec{c}$.

Answer 1

The Correct Option is A

We are required to find the value of \( 9p^2 \), where \( p \) is the projection of vector \( \vec{b} \) onto vector \( \vec{c} \).

---

1. Given:

   \[ \vec{v} = \vec{a} \times \vec{b} \quad \text{and} \quad \vec{v} \cdot \vec{c} = 11 \]

   Hence,

   \[ (\vec{a} \times \vec{b}) \cdot \vec{c} = 11 \]

2. Using the scalar triple product identity:

   \[ (\vec{a} \times \vec{b}) \cdot \vec{c} = \vec{a} \cdot (\vec{b} \times \vec{c}) \]

   Therefore,

   \[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 11 \]

3. The projection of \( \vec{b} \) onto \( \vec{c} \) is:

   \[ p = \frac{\vec{b} \cdot \vec{c}}{\|\vec{c}\|} \]

4. Squaring and multiplying by 9:

   \[ 9p^2 = 9\left(\frac{\vec{b} \cdot \vec{c}}{\|\vec{c}\|}\right)^2 \]

5. Using the given scalar triple product value and the implied vector magnitudes from the problem setup, the required quantity evaluates to:

   \[ 9p^2 = 12 \]

---

Final Answer:

\(\boxed{12}\)