The equation of the plane which passes through the line of intersection of planes ` vec rdot vec n_1=, q_1, vec rdot vec n_2=q_2` and the is parallel to the line of intersection of planers ` vec rdot vec n_3=q_3a n d vec rdot vec n_4-q_4` is a. `b . c.[d . e. n_f .2g . h. , i. n_j .3k . l. , m. n_n .4o.p .](q . r. rdots . n_t .1u . v.-w . q_x .1y . z. aa.)=[b b . cc. n_d d .1e e . ff. , gg. n_h h .3i i . jj. , kk. n_l l .4m m . nn.](o o . pp. rdotq q . n_r r .2s s . tt.-u u . q_v v .2w w . xx. yy.)z z .` aaa. bbb. `c c c . ddd.[e e e . fff. n_g g g .1h h h . iii. , jjj. n_k k k .2l l l . mmm. , nnn. n_o o o .3p p p . qqq.](r r r . sss. rdott t t . n_u u u .4v v v . www.-x x x . q_y y y .4z z z . aaaa. bbbb.)=[c c c c . dddd. n_e e e e .4f f f f . gggg. , hhhh. n_i i i i .3j j j j . kkkk. , llll. n_m m m m .1n n n n . oooo.](p p p p . qqqq. rdotr r r r . n_s s s s .2t t t t . uuuu.-v v v v . q_w w w w .2x x x x . yyyy. zzzz.)a a a a a .` bbbbb. ccccc. `d d d d d . eeeee.[f f f f f . ggggg. n_h h h h h .4i i i i i . jjjjj. , kkkkk. n_l l l l l .3m m m m m . nnnnn. , ooooo. n_p p p p p .1q q q q q . rrrrr.](s s s s s . ttttt. rdotu u u u u . n_v v v v v .4w w w w w . xxxxx.-y y y y y . q_z z z z z .4a a a a a a . bbbbbb. cccccc.)=[d d d d d d . eeeeee. n_f f f f f f .1g g g g g g . hhhhhh. , iiiiii. n_j j j j j j .2k k k k k k . llllll. , mmmmmm. n_n n n n n n .3o o o o o o . pppppp.](q q q q q q . rrrrrr. rdots s s s s s . n_t t t t t t .2u u u u u u . vvvvvv.-w w w w w w . q_x x x x x x .2y y y y y y . zzzzzz. aaaaaaa.)b b b b b b b .` ccccccc. ddddddd. none of these

Options

A. `[vecn_(2)vecn_(3)vecn_(4)](vecr.vecn_(1)-q_(1))=[vecn_(1)vecn_(3)vecn_(4)](vecr.vecn_(2)-q_(2))`
B. `[vecn_(1)vecn_(2)vecn_(3)](vecr.vecn_(4)-q_(4))=[vecn_(4)vecn_(3)vecn_(1)](vecr.vecn_(2)-q_(2))`
C. `[vecn_(4)vecn_(3)vecn_(1)](vecr.vecn_(4)-q_(4))=[vecn_(1)vecn_(2)vecn_(3)](vecr.vecn_(2)-q_(2))`
D. none of these

Answer

a

Explanation

`vecr*vec(n_(1))+lamdavecr*vec(n_(2))= q_(1)+ lamdaq_(2)" "` (i)
where `lamda` is a parameter.
So `vec(n_(1))+ lamda vec(n_2)` is normal to plane (i). Now, any plane parallel to the line of intersection of the planes `vecr`.
`vec(n_3)= q_(3) and vecr*vec(n_(4)) = q_(4)` is of form `vecr*(vec(n_(3))xxvec(n_(4)))=d`. Hence, we must have
`" "[vec(n_1)+lamdavec(n_2)]*[vec(n_3)xxvec(n_4)]=0`
or `" "[vec(n_1)vec(n_3)vec(n_4)] + lamda[vec(n_2)vec(n_3)vec(n_4)]=0`
or `" " lamda = (-[vec(n_1)vec(n_2)vec(n_4)])/([vec(n_1)vec(n_3)vec(n_4)] )`
On putting this value in Eq. (i), we have the equation of the required plane as
`" "vecr*vec(n_1)-q_1= ([vec(n_1)vec(n_3)vec(n_4)])/([vec(n_2)vec(n_3)vec(n_4)])(r*vec(n_2)-q_2)`
or `" "[vec(n_2) vec(n_3)vec(n_4)](vecr*vec(n_1)-q_1)`
`" "=[vec(n_1)vec(n_3)vec(n_4)] (vecr*vec(n_2) - q_2) `