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Statement 1: The direction cosines of...

Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as `l_1,m_1, n_1a n dl_2, m_2, n_2` are proportional to `l_1+l_2,m_1+m_2, n_1+n_2dot` Statement 2: The angle between the two intersection lines having direction cosines as `l_1,m_1, n_1a n dl_2, m_2, n_2` is given by `costheta=l_1l_2+m_1m_2+n_1n_2dot`

A

Both the statements are true, and Statement 2 is the correct explanation for Statement 1.

B

Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.

C

Statement 1 is true and Statement 2 is false.

D

Statement 1 is false and Statement 2 is true.

Text Solution

Verified by Experts

The correct Answer is:
B

We know that vector in the direction of angular bisector of unit vectors `veca and vecb` is `(veca + vecb)/(2 cos( theta//2))`
where `veca = vec(AB) =l_1hati + m_1hatj +n_1hatk`
and `vecb = vec(AD) = l_2 hati +m_2 hatj + n_3 hatk`
Thus, unit vector along the bisector is
`(l_1 +l_)/(cos (theta/2)) hati + (m_1 + m_2) /( cos (theta//2)) hatj + (n_1 + n_2)/(cos (theta//2)) hatk`
Hence, Statement 1 is true.
Also, in triangle ABD, by cosines rule
`cos theta = (AB^(2) + AD^(2) - BD^(2))/(2AB * AD)`
`=(1+1 -|(l_1 -l_2)hati + (m_1- m_2)hatj + (n_1- n_2)hatk|^(2))/(2)`
`= (2 - [(l_1 -l_2)^(2) + (m_1-m_2)^(2) + (n_1-n_2)^(2)])/(2)`
`= (2-[2-2(l_1l_2 +m_1m_2 + n_1n_2)])/(2)`
`= l_1l_2 + m_1m_2 + n_1n_2`
Hence, Statment 2 is true but does not explain Statement 1.
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