Find the equation of plane which is at a distance `4/(sqrt(14))` from the origin and is normal to vector `2 hat i+ hat j-3 hat kdot`

Find the direction cosines of the vector hat i+2 hat j+3 hat kdot

The equation of the plane at a distance P from the origin and perpendicular to the unit normal vector hat(d) is

Find the equation of the sphere described on the joint of points Aa n dB having position vectors 2 hat i+6 hat j-7 hat ka n d-2 hat i+4 hat j-3 hat k , respectively, as the diameter. Find the center and the radius of the sphere.

Find the equation of the plane which is parallel to the lines vec r= hat i+ hat j+lambda(2 hat i+ hat j+4 hat k)a n d(x+1)/(-3)=(y-3)/2=(z+2)/1"" and is passing through the point (0,1,-1 ).

Find the equation the plane which contain the line of intersection of the planes vec rdot( hat i+2 hat j+3 hat k)-4=0a n d vec rdot(2 hat i+ hat j- hat k)+5=0 and which is perpendicular to the plane vec r(5 hat i+3 hat j-6 hat k)+8=0 .

Find the projection of the vector hat i + hat j on the vector hat i - hat j .

Find the vector of length 3 unit which is perpendicular to hat i+ hat j+ hat k and lies in the plane of hat i+ hat j+ hat ka n d2 hat k-3 hat j .

Find the equation of the plane passing through (3,4,-1), which is parallel to the plane vec rdot(2 hat i-3 hat j+5 hat k)+7=0.

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)