Question

Given that \( \vec{a} = 2\hat{i} - \lambda \hat{j} + 5\hat{k} \), \( \vec{b} = \mu \hat{i} + 7\hat{j} + 3\hat{k} \), and the midpoint of \( \overrightarrow{AB} = 3\hat{i} + 2\hat{j} + 4\hat{k} \), find \( \lambda + \mu \).

Hint:

To find the midpoint of two vectors, use the formula \( \frac{\vec{a} + \vec{b}}{2} \).

Answer 1

Step 1: Midpoint formula.
The midpoint of two vectors \( \vec{a} \) and \( \vec{b} \) is given by: \[ \text{Midpoint} = \frac{\vec{a} + \vec{b}}{2} \] We are given that the midpoint is \( 3\hat{i} + 2\hat{j} + 4\hat{k} \), so we have: \[ \frac{\vec{a} + \vec{b}}{2} = 3\hat{i} + 2\hat{j} + 4\hat{k} \] Multiplying both sides by 2: \[ \vec{a} + \vec{b} = 6\hat{i} + 4\hat{j} + 8\hat{k} \]
Step 2: Substitute the values of \( \vec{a} \) and \( \vec{b} \).
Substitute \( \vec{a} = 2\hat{i} - \lambda \hat{j} + 5\hat{k} \) and \( \vec{b} = \mu \hat{i} + 7\hat{j} + 3\hat{k} \) into the equation: \[ (2\hat{i} - \lambda \hat{j} + 5\hat{k}) + (\mu \hat{i} + 7\hat{j} + 3\hat{k}) = 6\hat{i} + 4\hat{j} + 8\hat{k} \]
Step 3: Combine like terms.
Simplifying the left-hand side: \[ (2 + \mu)\hat{i} + (-\lambda + 7)\hat{j} + (5 + 3)\hat{k} = 6\hat{i} + 4\hat{j} + 8\hat{k} \]
Step 4: Equate the components.
Equating the components of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \): \[ 2 + \mu = 6, \quad -\lambda + 7 = 4, \quad 5 + 3 = 8 \]
Step 5: Solve for \( \lambda \) and \( \mu \).
From \( 2 + \mu = 6 \), we get: \[ \mu = 4 \] From \( -\lambda + 7 = 4 \), we get: \[ \lambda = 3 \]
Step 6: Find \( \lambda + \mu \).
Thus: \[ \lambda + \mu = 3 + 4 = 7 \]