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

determine whether the given planes are parallel or perpendicular,and in case they are neither, find the angles between them. (a)7x+5y+6z+30=0 and 3x-y-10z+4=0 

(b)2x+y+3z-2=0 and x-2y+5=0 

(c)2x-2y+4z+5=0 and 3x-3y+6z-1=0 

(d)2x-y+3z-1=0 and 2x-y+3z+3=0 

(e)4x+8y+z-8=0 and y+z-4=0

Find the vector equation of the line passing through(1, 2, 3)and parallel to the planes \(\vec r.=(\hat i-\hat j+2\hat k)=5\) and \(\vec r.(3\hat i+\hat j+\hat k)=6\).

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

If the points (1, 1, p) and (-3, 0, 1) be equidistant from the plane  \(\vec r(3\hat i+4\hat j-12\hat k)+13=0,\) then find the value of p.

In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(a) (0,0,0)      3x-4y+12z=3
(b) (3,-2,1)    2x-y+2z+3=0
(c) (2,3,-5)     x+2y-2z=9
(d) (-6,0,0)     2x-3y+6z-2=0

Find the shortest distance between the lines whose vector equations are

\(\overrightarrow r=(1-t)\hat i+(t-2)\hat j+(3-2t)\hat k\) and

\(\overrightarrow r=(s+1)\hat i+(2s-1)\hat j-(2s+1)\hat k\)

Find the equation of the plane passing through the line of intersection of the planes \(\vec r.(\hat i+\hat j+\hat k)=1 \) and \(\vec r.(2\hat i+3\hat j-\hat k)+4=0\) and parallel to x-axis.

If O be the origin and the coordinates of P be (1, 2, -3),then find the equation of the plane passing through P and perpendicular to OP.

Find the shortest distance between the lines whose vector equations are

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

and \(\overrightarrow r=(4\hat i+5\hat j+6\hat k)+\mu(2\hat i+3\hat j+\hat k)\)

Find the shortest distance between the lines  \(\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}\) and \(\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}\)

Find the shortest distance between the lines
\(\overrightarrow r=(\hat i+2\hat j+\hat k)+\lambda(\hat i-\hat j+\hat k)\) and
\(\overrightarrow r=2\hat i-\hat j-\hat k+\mu\,(2\hat i+2\hat j+2\hat k)\)

Find the vector equation of the line passing through the point (1, 2, -4)and perpendicular to the two lines: \(\frac {x-8}{3}=\frac {y+19}{-16}=\frac {z-10}{7}\) and \(\frac {x-15}{3}=\frac {y-29}{8} =\frac {z-5}{-5}\)

If l₁,m₁,n₁ and l₂,m₂,n₂ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m₁n₂-m₂n₁,n₁l₂-n₂l₁,l₁m₂-l₂m₁.

Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then \(\frac {1}{a^2}+\frac {1}{b^2}+\frac {1}{c^2} =\frac {1}{p^2}\).

Distance between the two planes: 2x+3y+4z = 4 and 4x+6y+8z = 12 is

\(The\  planes: 2x-y+4z=5 \ and \ 5x-2.5y+10z=6\  are\)

Find the angle between the planes whose vector equations are 

\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=5\) and \(\overrightarrow r.(3\hat i-3\hat j+5\hat k)=3\)

Find the equation of the plane through the line of intersection of the planes
x+y+z=1 and 2x+3y+4z=5 which is perpendicular to the plane x-y+z= 0

Find the vector equation of the plane passing through the intersection of the planes

\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=7\),\(\overrightarrow r.(2\hat i+5\hat j+3\hat k)=9\) and through the point ( 2, 1, 3 ).