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Let P1 denote the equation of a plane to...

Let `P_1` denote the equation of a plane to which the vector `( hat i+ hat j)` is normal and which contais the line whose equation is ` vec r= hat i+ hat j+ hat k+lambda( hat i- hat j- hat k)a n dP_2` denote the equation of the plane containing the line `L` and a point with position vector ` hat jdot` Which of the following holds good? a. The equation of `b . c. d. P_e .1f . g. i sx+y=2. h.` i. j. The equation of `k . l. m. P_n .2o.p . i s(q . r. hat i t. -2 hat j x. + hat k bb. )=2. ff.` gg. hh. The acute angle between `i i . jj. kk. P_l l .1m m . nn. a n do o . P_p p .2q q . rr.i s ss. tt.cot^(u u . vv.-1w w .)x x .(y y . zz.sqrt(a a a .3b b b .)c c c . ddd.)dote e e .` fff. ggg. The angle between the plane `h h h . iii. jjj. P_k k k .2l l l . mmm.n n n .` ooo. and the line `p p p . qqq. Li s rrr. sss.tan^(t t t . uuu.-1v v v .)w w w .sqrt(x x x .3y y y .)z z z .dota a a a .` bbbb.

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Let P_1 denote the equation of a plane to which the vector ( hat i+ hat j) is normal and which contais the line whose equation is vec r= hat i+ hat j+ hat k+lambda( hat i- hat j- hat k)a n dP_2 denote the equation of the plane containing the line L and a point with position vector hat jdot Which of the following holds good? a. The equation of P_1 is x+y=2. b. The equation of P_2 is vec(r) . (i-2j+k) = 2 c. The acute angle between P_1 and P_2 is cot^(-1) sqrt(3) d. The angle between plane P_2 and the line L is tan^(-1) sqrt(3)

Show that the plane whose vector equation is vec r*( hat i+2 hat j= hat k)=3 contains the line whose vector equation is vec r*( hat i+ hat j)+lambda(2 hat i+ hat j+4 hat k)dot

Show that the plane whose vector equation is vec rdot( hat i+2 hat j- hat k)=1. and the line whose vector equation is vec r=(- hat i+ hat j+ hat k)+lambda(2 hat i+ hat j+4 hat k) are parallel. Also, find the distance between them.

Find the equation of the line passing through the point hat i+ hat j- hat k and perpendicular to the lines vec r= hat i+lambda(2 hat i+ hat j-3 hat k)a n d vec r=(2 hat i+ hat j- hat k)dot

Write the cartesian equation of the following line given in vector form : vec r=2 hat i+ hat j-4 hat k+lambda\ ( hat i- hat j- hat k)

Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2 hat i- hat j+4 hat k and is in the direction hat i+2 hat j- hat k .

Show that the lines vec r=( hat i+ hat j- hat k)+lambda(3 hat i- hat j)a n d vec r=(4 hat i- hat k)+mu(2 hat i+3 hat k) are coplanar. Also, find the plane containing these two lines.

Find the angle between the lines vec r= dot( hat i+2 hat j- hat k)+lambda( hat i- hat j+ hat k) and the plane vec r.(2 hat i- hat j+ hat k)=4.

The distance between the line vec r=(2 hat i-2 hat j+3 hat k)+lambda( hat i- hat j+4 hat k) and plane vec rdot( hat i+5 hat j+ hat k)=5.

Find the equation of the plane through the point 2 hat i+ hat j- hat k and passing through the line of intersection of the plane vec rdot(( hat i+3 hat j- hat k)=0\ a n d\ vec rdot(( hat j+2 hat k)=0.

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