If the sides $AB$ and $AC$ of $\triangle ABC$ are represented by vectors $\hat{i} + \hat{j} + 4 \hat{k}$ and $3 \hat{i} - \hat{j} + 4 \hat{k}$ respectively, then the length of the median through A on BC is :
Show Hint
To find the length of a median, find the midpoint of the opposite side and calculate the distance between the vertex and the midpoint.
The length of the median is the distance from A to the midpoint of BC. First, we find the midpoint of BC, which is the average of the coordinates of points B and C. Let the coordinates of B and C be $\mathbf{B} = (1, 1, 4)$ and $\mathbf{C} = (3, -1, 4)$ respectively. The midpoint $\mathbf{M}$ is:
\[
\mathbf{M} = \left( \frac{1 + 3}{2}, \frac{1 + (-1)}{2}, \frac{4 + 4}{2} \right) = (2, 0, 4)
\]
Now, the distance from A $(\hat{i} + \hat{j} + 4\hat{k})$ to M $(2, 0, 4)$ is:
\[
d = \sqrt{(2 - 1)^2 + (0 - 1)^2 + (4 - 4)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{2}
\]
Thus, the length of the median is $\frac{\sqrt{34}}{2}$.
