The vectors overline{AB} = 3hat{i} + 4hat{k} | vedclass

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(D) Let $D$ be the midpoint of side $BC$. The median from vertex $A$ is represented by the vector $\overline{AD}$.By the property of the median in a t

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$\sqrt{33}$

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(D) Let $D$ be the midpoint of side $BC$. The median from vertex $A$ is represented by the vector $\overline{AD}$.By the property of the median in a triangle,$\overline{AD} = \frac{\overline{AB} + \overline{AC}}{2}$.Substituting the given vectors: $\overline{AD} = \frac{(3\hat{i} + 4\hat{k}) + (5\hat{i} - 2\hat{j} + 4\hat{k})}{2}$.$\overline{AD} = \frac{8\hat{i} - 2\hat{j} + 8\hat{k}}{2} = 4\hat{i} - \hat{j} + 4\hat{k}$.The length of the median is the magnitude of vector $\overline{AD}$:$|\overline{AD}| = \sqrt{4^2 + (-1)^2 + 4^2} = \sqrt{16 + 1 + 16} = \sqrt{33}$ units.

The vectors $\overline{AB} = 3\hat{i} + 4\hat{k}$ and $\overline{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ represent the sides of a triangle $ABC$. What is the length of the median passing through vertex $A$?

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