Discussion Forum

 If the vectors AB= $3\hat{i}+4\hat{k}$  and $AC= 5\hat{i}-2\hat{j}+4\hat{k}$  are the sides of $\triangle$ABC  , then the length of the median through A is

Answer:

Explanation:

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

$\therefore$   AB+BC+CA=0

  $\Rightarrow$   BC=AC-AB

 $\Rightarrow$      $BM=\frac{AC-AB}{2}$

   ($\because$  M is a mid point of BC )

Also, AB+BM+MA=0

                  (By properties of a triangle)

   $\Rightarrow$     $AB+\frac{AC-AB}{2}=AM$

$\Rightarrow$     $AM=\frac{AB+AC}{2}$

      = $\frac{3\hat{i}+4\hat{k}+5\hat{i}-2\hat{k}+4\hat{k}}{2}$

        =  $4\hat{i}-\hat{j}+4\hat{k}$

 $\Rightarrow$|AM|=   $\sqrt{4^{2}+1^{2}+4^{2}}=\sqrt{33}$