# If the coordinates of the points A,B,C be (-1,3,2),(2,3,5) and (3,5,-2) respectively, then $\angle A=$ Option 1) 0 Option 2) $45^{\circ}$ Option 3) $60^{\circ}$ Option 4) $90^{\circ}$

As we discussed in concept

Ange between two lines in terms of direction cosines and direction ratios -

If two lines having direction ratios a1,b1,c1 and a2,b2,c2 then the angle between them is given by

$\cos \Theta = \frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a{_{1}}^{2}+b{_{1}}^{2}+c{_{1}}^{2}}\sqrt{a{_{2}}^{2}+b{_{2}}^{2}+c{_{2}}^{2}}}$

If two lines have direction ratios as l1,m1,n1 and l2,m2,n2 then the angle betwee them is given by

$\cos \Theta = l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}$

-

Direction ratio of AB = (3,0,3)

Direction ratio of AC = (4,2,-4)

So, Cos A = $\frac{3\times 4+0\times 2\:+3\times -4}{\sqrt{9+0+9}\times \sqrt{16+4+16}}$

= 0

$A\:=\:90^{\circ}$

Option 1)

0

This option is incorrect.

Option 2)

$45^{\circ}$

This option is incorrect.

Option 3)

$60^{\circ}$

This option is incorrect.

Option 4)

$90^{\circ}$

This option is correct.

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