Question

The position vectors of the points $A$ and $B$ are $\vec{a} = 2\hat{i} - \lambda \hat{j} + 5\hat{k}$ and $\vec{b} = \mu \hat{i} + 7\hat{j} + 3\hat{k}$ respectively. If the position vector of the mid-point of the line segment $AB$ is $\vec{c} = 3\hat{i} + 2\hat{j} + 4\hat{k}$, then the value of $\lambda + \mu$ is equal to

• A. 6
• B. 7
• C. 8
• D. 9
• E. 10

Hint:

For midpoint vector problems, simply add corresponding components of the endpoint vectors and divide by two. It is identical to the midpoint formula in coordinate geometry.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The position vector of the mid-point of a line segment connecting two points with position vectors \( \vec{a} \) and \( \vec{b} \) is \( \vec{c} = \frac{\vec{a} + \vec{b}}{2} \).

Step 2: Detailed Explanation:
\[ \vec{c} = \frac{(2\hat{i} - \lambda \hat{j} + 5\hat{k}) + (\mu \hat{i} + 7\hat{j} + 3\hat{k})}{2} \]
\[ \vec{c} = \frac{2 + \mu}{2}\hat{i} + \frac{7 - \lambda}{2}\hat{j} + \frac{8}{2}\hat{k} \]
Given \( \vec{c} = 3\hat{i} + 2\hat{j} + 4\hat{k} \).
Comparing components:
\( \hat{i} \) component: \( \frac{2 + \mu}{2} = 3 \implies 2 + \mu = 6 \implies \mu = 4 \)
\( \hat{j} \) component: \( \frac{7 - \lambda}{2} = 2 \implies 7 - \lambda = 4 \implies \lambda = 3 \)
Calculating \( \lambda + \mu \):
\[ 3 + 4 = 7 \]

Step 3: Final Answer:
The value of \( \lambda + \mu \) is 7.