Question

If $\overrightarrow{AB}=\hat{j}+\hat{k}$ and $\overrightarrow{AC}=3\hat{i}-\hat{j}+4\hat{k}$ represent the two vectors along the sides $AB$ and $AC$ of $\triangle ABC$, prove that the median $\overrightarrow{AD}=\dfrac{\overrightarrow{AB}+\overrightarrow{AC}}{2}$ where $D$ is the midpoint of $BC$. Hence, find the length of the median $AD$.

Hint:

In vector geometry, the midpoint vector is always the average of the two endpoint vectors. Hence, the median from one vertex of a triangle is half the sum of the two side vectors starting from that vertex.

Answer 1

Step 1: Write the given vectors.
The vectors along the sides of triangle are given as \[ \overrightarrow{AB}=\hat{j}+\hat{k} \] \[ \overrightarrow{AC}=3\hat{i}-\hat{j}+4\hat{k} \] Step 2: Express vectors in component form.
\[ \overrightarrow{AB}=0\hat{i}+1\hat{j}+1\hat{k} \] \[ \overrightarrow{AC}=3\hat{i}-1\hat{j}+4\hat{k} \] Step 3: Use the formula of median in vector form.
If $D$ is the midpoint of $BC$, then the median from $A$ to $BC$ is given by \[ \overrightarrow{AD}=\frac{\overrightarrow{AB}+\overrightarrow{AC}}{2} \] Step 4: Substitute the given vectors.
\[ \overrightarrow{AD}=\frac{(0\hat{i}+1\hat{j}+1\hat{k})+(3\hat{i}-1\hat{j}+4\hat{k})}{2} \] Add the vectors: \[ = \frac{(3\hat{i}+0\hat{j}+5\hat{k})}{2} \] Thus \[ \overrightarrow{AD}=\frac{3}{2}\hat{i}+\frac{5}{2}\hat{k} \] Step 5: Find the magnitude of median $AD$.
\[ |\overrightarrow{AD}|=\sqrt{\left(\frac{3}{2}\right)^2+0^2+\left(\frac{5}{2}\right)^2} \] \[ =\sqrt{\frac{9}{4}+\frac{25}{4}} \] \[ =\sqrt{\frac{34}{4}} \] \[ =\frac{\sqrt{34}}{2} \] Step 6: Final conclusion.
Thus the vector of the median is \[ \overrightarrow{AD}=\frac{\overrightarrow{AB}+\overrightarrow{AC}}{2} \] and the length of the median is \[ \frac{\sqrt{34}}{2} \] Final Answer: \[ |\overrightarrow{AD}|=\boxed{\frac{\sqrt{34}}{2}} \]