Question

The vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a $\triangle ABC$. The length of the median through $A$ is:

• A. $\sqrt{18}$
• B. $\sqrt{72}$
• C. $\sqrt{33}$
• D. $\sqrt{288}$

Answer 1

The Correct Option is C

1. Find the midpoint vector:

Given vectors \( \overrightarrow{AB} = 3\hat{i} + 4\hat{k} \) and \( \overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k} \).

The median through \( A \) is the vector from \( A \) to the midpoint of \( BC \):

\[ \overrightarrow{AM} = \frac{\overrightarrow{AB} + \overrightarrow{AC}}{2} = \frac{(3\hat{i} + 4\hat{k}) + (5\hat{i} - 2\hat{j} + 4\hat{k})}{2} = 4\hat{i} - \hat{j} + 4\hat{k} \]

2. Compute the length of the median:

\[ |\overrightarrow{AM}| = \sqrt{4^2 + (-1)^2 + 4^2} = \sqrt{16 + 1 + 16} = \sqrt{33} \]

Correct Answer: (C) \( \sqrt{33} \)

Answer 2

The median from vertex $A$ to the midpoint $M$ of side $BC$ is given by: \[ \overrightarrow{AM} = \frac{\overrightarrow{AB} + \overrightarrow{AC}}{2}. \] Compute the median vector: \[ \overrightarrow{AM} = \frac{(3\hat{i} + 4\hat{k}) + (5\hat{i} - 2\hat{j} + 4\hat{k})}{2} = \frac{8\hat{i} - 2\hat{j} + 8\hat{k}}{2} = 4\hat{i} - \hat{j} + 4\hat{k}. \] Calculate the length of the median: \[ |\overrightarrow{AM}| = \sqrt{4^2 + (-1)^2 + 4^2} = \sqrt{16 + 1 + 16} = \sqrt{33}. \] Hence, the length of the median through $A$ is $\sqrt{33}$.