The two vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) represent the two sides \(\overline{A B}\) and \(\overline{A C}\) respectively of a \(\triangle A B C\). The length of the median through \(A\) is

We know that, the sum of three vectors of triangle is zero.



\(\begin{array}{ll}\therefore & \mathbf{A B}+\mathbf{B C}+\mathbf{C A}=0 \\ \Rightarrow \quad & \mathbf{B C}=\mathbf{A C}-\mathbf{A B} \\ \Rightarrow \mathbf{B} \mathbf{M}= & \frac{\mathbf{A C}-\mathbf{A B}}{2}(\text { since, } M \text { is a mid-point of } B C)\end{array}\)
And also, \(\mathbf{A B}+\mathbf{B} \mathbf{M}+\mathbf{M A}=0\)
\(\Rightarrow \quad \mathbf{A B}+\frac{\mathbf{A C}-\mathbf{A B}}{2}=\mathbf{A M}\)
\(\Rightarrow \quad \mathbf{A M}=\frac{\mathbf{A B}+\mathbf{A C}}{2}\)
\(\Rightarrow \quad \mathbf{A M}=\frac{(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})+(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})}{2}\)
\(\Rightarrow \quad \mathbf{A M}=\frac{2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}}{2}\)
\(\Rightarrow \quad \mathbf{A M}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\)
\(\therefore \quad|\mathbf{A M}|=\sqrt{(1)^{2}+(2)^{2}+(3)^{2}}\)
\(=\sqrt{1+4+9}=\sqrt{14}\)