Question

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

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

Hint:

To make vector calculations simpler, you can assign vertex A as the origin $(0,0,0)$. This makes the coordinates of point B $(3,0,4)$ and point C $(5,-2,4)$. Finding the midpoint of BC directly yields position D $(4,-1,4)$, and the distance formula gives $\sqrt{4^2+(-1)^2+4^2} = \sqrt{33}$!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given two position vectors representing sides $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$ of a triangle ABC. We need to compute the mathematical magnitude (length) of the median vector originating from vertex A and terminating at the midpoint of side BC.

Step 2: Key Formula or Approach:
Let D be the midpoint of side BC. By vector geometry rules, the median vector $\overrightarrow{\text{AD}}$ pointing from vertex A to the midpoint of the opposite side BC is given by the average of the two adjacent side vectors:
$$\overrightarrow{\text{AD}} = \frac{\overrightarrow{\text{AB}} + \overrightarrow{\text{AC}}}{2}$$

Step 3: Detailed Explanation:
Let's find the sum of the two given side vectors:
$$\overrightarrow{\text{AB}} = 3\hat{\text{i}} + 0\hat{\text{j}} + 4\hat{\text{k}}$$ $$\overrightarrow{\text{AC}} = 5\hat{\text{i}} - 2\hat{\text{j}} + 4\hat{\text{k}}$$ Add their corresponding components together:
$$\overrightarrow{\text{AB}} + \overrightarrow{\text{AC}} = (3 + 5)\hat{\text{i}} + (0 - 2)\hat{\text{j}} + (4 + 4)\hat{\text{k}} = 8\hat{\text{i}} - 2\hat{\text{j}} + 8\hat{\text{k}}$$ Now, calculate the median vector $\overrightarrow{\text{AD}}$ by dividing this sum by 2:
$$\overrightarrow{\text{AD}} = \frac{8\hat{\text{i}} - 2\hat{\text{j}} + 8\hat{\text{k}}}{2} = 4\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$$ To find the final length of this median, compute the vector magnitude of $\overrightarrow{\text{AD}}$:
$$\left| \overrightarrow{\text{AD}} \right| = \sqrt{(4)^2 + (-1)^2 + (4)^2}$$ $$\left| \overrightarrow{\text{AD}} \right| = \sqrt{16 + 1 + 16} = \sqrt{33}\text{ units}$$ This matches perfectly with option (A).

Step 4: Final Answer:
The length of the median through A is $\sqrt{33}$ units, which corresponds to option (A).