Question

If \(\overrightarrow{a}=\hat{i}+\lambda\hat{j}-2\hat{k}, \overrightarrow{b}=2\hat{i}-3\hat{j}+5\hat{k}\ and\ \overrightarrow{a}\cdot\overrightarrow{b}=-20\), then the value of \(\lambda\) is equal to

• A. 2
• B. -2
• C. -4
• D. 4
• E. 5

Answer 1

The Correct Option is D

Step 1: Use the dot product formula 
The dot product of two vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \) is given by: \[ \overrightarrow{a} \cdot \overrightarrow{b} = a_x b_x + a_y b_y + a_z b_z \] Given: \[ \overrightarrow{a} = \hat{i} + \lambda\hat{j} - 2\hat{k}, \quad \overrightarrow{b} = 2\hat{i} - 3\hat{j} + 5\hat{k} \] Also, we are given: \[ \overrightarrow{a} \cdot \overrightarrow{b} = -20 \]

Step 2: Compute the dot product 
Expanding using the formula: \[ (1 \times 2) + (\lambda \times -3) + (-2 \times 5) = -20 \] \[ 2 - 3\lambda - 10 = -20 \] \[ -3\lambda - 8 = -20 \] \[ -3\lambda = -12 \] \[ \lambda = 4 \]

Final Answer: \( \lambda \) is 4.