The angle b/w the lines joinong the points (1,1,0) , (2,-1,2) , and (1,1,1) , (3,2,0) is

a. cos^-1(2/9)

b. 90^o

c. 0^o

d. cos^-1[2(2)^1/2/9]

Question

The angle b/w the lines joinong the points (1,1,0) , (2,-1,2) ,
and (1,1,1) , (3,2,0) is
a cos-1(2/9)
b 90o
c 0o
d cos-1[2(2)1/2/9]

Ajanta Trivedi · Accepted Answer

let the given points are $A (1, 1, 0), B (2, - 1, 2), C (1, 1, 1) a n d D (3, 2, 0)$

the direction ratio of the line passing through the points A and B are $1 - 2, 1 + 1, 0 - 2$

i.e. direction ratio of line AB are $(- 1, 2, - 2)$

let $a_{1} = - 1, b_{1} = 2 a n d c_{1} = - 2$ .

the direction ratios of the line CD are $3 - 1, 2 - 1, 0 - 1$

i.e. $(2, 1, - 1)$ .

let $a_{2} = 2, b_{2} = 1 a n d c_{2} = - 1$ .

the angle between two lines whose direction ratios are proportional to $a_{1}, b_{1}, c_{1} a n d a_{2}, b_{2}, c_{2}$

respectively is given by:

$\cos θ = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{{a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}} \cdot \sqrt{{a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}}}$

therefore the angle between the lines AB and CD is

$\cos θ = \frac{- 1 * 2 + 2 * 1 + (- 2) * (- 1)}{\sqrt{1^{2} + 2^{2} + 2^{2}} \cdot \sqrt{2^{2} + 1^{2} + (- 1)^{2}}} \cos θ = \frac{- 2 + 2 + 2}{\sqrt{9} \cdot \sqrt{9}} = \frac{2}{9}$

$\Rightarrow θ = \cos^{- 1} (\frac{2}{9})$

hope this helps you.

cheers!!

The angle b/w the lines joinong the points (1,1,0) , (2,-1,2) , and (1,1,1) , (3,2,0) is a. cos-1(2/9) b. 90o c. 0o d. cos-1[2(2)1/2/9]

The angle b/w the lines joinong the points (1,1,0) , (2,-1,2) , and (1,1,1) , (3,2,0) is

a. cos^-1(2/9)

b. 90^o

c. 0^o

d. cos^-1[2(2)^1/2/9]