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

Image of the point P with position vector 7hat i-hat j+2hat k in the line whose vector equation is,vec r=9hat i+5hat j+5hat k+lambda(hat i+3hat j+5hat k) has the position vector

Image of the point P with position vector 7hat(i) - hat(j) + 2hat(k) in the line whose vector equation is vec(r) = 9hat(i) + 5hat(j) + 5hat(k) + lambda(hat(i) + 3hat(j) + 5hat(k)) has the position .

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=2hat i+hat j-4hat k+lambda(hat i-hat j-hat k)

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

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

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.

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

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