List of top Mathematics Questions on Three Dimensional Geometry asked in CBSE CLASS XII

Find the vector and cartesian equation of the planes 

(a) that passes through the point (1,0,-2)and the normal to the plane is \(\hat i+\hat j-\hat k\). 

(b) that passes through the point(1,4,6)and the normal vector to the plane is \(\hat i-2\hat j+\hat k\).

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x+2y+3z=5 and 3x+3y+z = 0.

Find the angles between the following pairs of lines:
(i) \(\overrightarrow r= 2\hat i-5\hat j+\hat k+ \lambda (3\hat i-2\hat j+6\hat k)\)and

\(\overrightarrow r=7\hat i-6\hat k+\mu(\hat i+2\hat j+2\hat k)\)

(ii) \(\overrightarrow r=3\hat i+\hat j-2\hat k+\lambda(\hat i-\hat j-2\hat k)\) and

\(\overrightarrow r = 2\hat i-\hat j-56\hat k+\mu(3\hat i-5\hat j-4\hat k)\)

Find the cartesian equations of the following planes:

\(\overrightarrow r.(\hat i+\hat j-\hat k)=2\) (b) \(\overrightarrow r.(2\hat i+3\hat j-4\hat k)=1\)

(c) \(\overrightarrow r.[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k)=15\)

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector \(3\hat i+5\hat j-6\hat k.\)

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin. 
(a) z=2  (b)  x+y+z=1 
(c)2x+3y-z=5  (d)  5y+8=0

Show that the lines\(\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}\) and \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\) are perpendicular to each other.

Find the values of P so the line\(\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2}\) and \(\frac{7-7x}{3p}=\frac{y-5}{1}=\frac{6-z}{5}\) are at right angles.