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 midpoint of a line segment connecting two points \(\vec{a}\) and \(\vec{b}\) is the average of their position vectors: \(\vec{c} = \frac{\vec{a} + \vec{b}}{2}\).
Step 2: Key Formula or Approach:
Equate the \(i\), \(j\), and \(k\) components of \(\frac{\vec{a} + \vec{b}}{2}\) to the corresponding components of \(\vec{c}\).
Step 3: Detailed Explanation:
The midpoint vector is:
\[ \frac{(2 + \mu)\hat{i} + (-\lambda + 7)\hat{j} + (5 + 3)\hat{k}}{2} = 3\hat{i} + 2\hat{j} + 4\hat{k} \] Equate the \(\hat{i}\) components:
\[ \frac{2 + \mu}{2} = 3 \implies 2 + \mu = 6 \implies \mu = 4 \] Equate the \(\hat{j}\) components:
\[ \frac{-\lambda + 7}{2} = 2 \implies -\lambda + 7 = 4 \implies \lambda = 3 \] Check the \(\hat{k}\) component just to be sure: \(\frac{5 + 3}{2} = 4\), which is correct.
We need to find \(\lambda + \mu\):
\[ \lambda + \mu = 3 + 4 = 7 \] Step 4: Final Answer:
The value of \(\lambda + \mu\) is 7.