PrepWiser › Learn › Mathematics › Three Dimensional GeometryThree Dimensional Geometry — JEE Main PYQs & Practice Questions Three Dimensional Geometry is a Mathematics chapter in the JEE Main syllabus. This page brings together 297 previous-year and practice questions on Three Dimensional Geometry, organised into 7 topics: Direction Cosines and Ratios, Equation of Line in 3D, Equation of Plane, Distance and Angle in 3D, Skew Lines and Shortest Distance, Angle Between Planes and Distance of Point from Plane. Most questions come with a step-by-step solution you can work through free in PrepWiser.
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Direction Cosines and Ratios Let P(x, y, z) be a point in the first octant, whose projection in the x y-plane is the… Let the direction cosines of two lines satisfy the equations : 4l+m-n=0 and… Let a unit vector O P make angles α, β, γ with the positive directions of the… Let α be the angle between the lines whose direction cosines satisfy the equations l + m… A vector v⃗ in the first octant is inclined to the x-axis at 60°, to the y-axis at 45… Let a unit vector u=x î+y ĵ+z k̂ make angles π/2, π/3 and (2 π)/3 with the vectors 1/√2… Each of the angles β and γ that a given line makes with the positive y - and z-axes… If two straight lines whose direction cosines are given by the relations l + m - n = 0… The angle between the straight lines, whose direction cosines are given by the equations… The point of intersection C of the plane 8 x+y+2 z=0 and the line joining the points… Let a line with direction ratios a,-4 a,-7 be perpendicular to the lines with direction… If the line of intersection of the planes a x+b y=3 and a x+b y+c z=0, a >0 makes an… Let P(-2,-1,1) and Q(56/17, 43/17, 111/17) be the vertices of the rhombus PRQS. If the… Let the image of the point (1,0,7) in the line x/1=(y-1)/2=(z-2)/3 be the point (α, β… Equation of Line in 3D Let the point A be the foot of perpendicular drawn from the point P(a,b,0) on the line… The square of the distance of the image of the point (6,1,5) in the line… The distance of the point (1, 1, 9) from the point of intersection of the line (x - 3)/1… Let (α, β, γ) be the image of the point (8,5,7) in the line (x-1)/2=(y+1)/3=(z-2)/5… The square of the distance of the point of intersection of the lines… Let (α, β, γ) be the foot of perpendicular from the point (1,2,3) on the line… Let the line L₁ be parallel to the vector -3î+2ĵ+4k̂ and pass through the point (2,6,7)… Let (α,β,γ) be the co-ordinates of the foot of the perpendicular drawn from the point… Let d be the distance of the point of intersection of the lines (x+6)/3=y/2=(z+1)/1 and… Let a line passing through the point (4,1,0) intersect the line L₁… Let L be the line (x+1)/2=(y+1)/3=(z+3)/6 and let S be the set of all points (a,b,c) on… The line, that is coplanar to the line (x+3)/-3=(y-1)/1=(z-5)/5, is : Let the vertices Q and R of the triangle PQR lie on the line (x+3)/5=(y-1)/2=(z+4)/3… Let A and B be two distinct points on the line L: (x-6)/3=(y-7)/2=(z-7)/-2. Both A and B… If the distance of the point P(43,α,β), β<0, from the line r⃗=4î-k̂+μ(2î+3k̂),μ∈ ℝ along… Let the image of the point P(1,6,a) in the line L:x/1=(y-1)/2=(z-a+1)/b,b>0, be… Let P(α, β, γ) be the image of the point Q(1,6,4) in the line x/1=(y-1)/2=(z-2)/3. Then… Let the line L: (x-1)/2=(y+1)/-1=(z-3)/1 intersect the plane 2 x+y+3 z=16 at the point… A plane P meets the coordinate axes at A, B and C respectively. The centroid of ΔABC is… The square of the distance of the point (-2,-8,6) from the line (x-1)/1=(y-1)/2=z/-1… Equation of Plane If (x, y, z) be an arbitrary point lying on a plane P which passes through the points… If the plane P passes through the intersection of two mutually perpendicular planes 2… A plane passing through the point (3, 1, 1) contains two lines whose direction ratios… The vector equation of the plane passing through the intersection of the planes r⃗ .( î… Let P₁:r⃗ . ( 2î + ĵ - 3k̂ ) = 4 be a plane. Let P₂ be another plane which passes… Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x - y + z +… Let P be a plane lx + my + nz = 0 containing the line, (1 - x)/1 = (y + 4)/2 = (z +… If the equation of the plane passing through the point (1,1,2) and perpendicular to the… The plane, passing through the points (0,-1,2) and (-1,2,1) and parallel to the line… The foot of perpendicular from the origin O to a plane P which meets the co-ordinate… Let the plane ax + by + cz = d pass through (2, 3, -5) and is perpendicular to the… The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right… Let P be the plane passing through the line (x-1)/1=(y-2)/-3=(z+5)/7 and the point… Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, -3, 1) and (2… Consider the line L given by the equation (x - 3)/2 = (y - 1)/1 = (z - 2)/1. Let Q be… Equation of a plane at a distance √(2/21) from the origin, which contains the line of… If the equation of the plane passing through the line of intersection of the planes 2x -… A plane P contains the line of intersection of the plane r⃗ ·(î+ĵ+k̂)=6 and r⃗ ·(2 î+3… Let the plane P pass through the intersection of the planes 2x+3y-z=2 and x+2y+3z=6, and… Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be… Distance and Angle in 3D If the distance of the point (1, -2, 3) from the plane x + 2y - 3z + 10 = 0 measured… Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x=y… A hall has a square floor of dimension 10 m × 10 m (see the figure) and vertical walls… The plane 2x-y+z=4 intersects the line segment joining the points A (a,-2,4) and B… The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to… Let the equation of plane passing through the line of intersection of the planes x+2 y+a… If the length of the perpendicular drawn from the point P(a, 4,2), a >0 on the line… Let λ₁, λ₂ be the values of λ for which the points (5/2, 1, λ) and (-2,0,1) are at equal… If the foot of the perpendicular from the point A(-1,4,3) on the plane P: 2 x+m y+n z=4… Let the line passing through the points P(2,-1,2) and Q(5,3,4) meet the plane x-y+z=4 at… Let the foot of perpendicular of the point P(3,-2,-9) on the plane passing through the… Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y +… Let P be an arbitrary point having sum of the squares of the distances from the planes x… Let Q be the foot of the perpendicular from the point P(7, -2, 13) on the plane… If the distance between the plane, 23x – 10y – 2z + 48 = 0 and the plane containing the… If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing… Let the line l: x=(1-y)/-2=(z-3)/λ, λ ∈ ℝ meet the plane P: x+2 y+3 z=4 at the point (α… The square of the distance of the point P(5,6,7) from the line (x-2)/2=(y-5)/3=(z-2)/4… Let the line L be the projection of the line (x - 1)/2 = (y - 3)/1 = (z - 4)/2 in the… The distance of the point (-1,9,-16) from the plane 2x+3y-z=5 measured parallel to the… Skew Lines and Shortest Distance The line l₁ passes through the point (2, 6, 2) and is perpendicular to the plane… If the shortest distance between the lines r⃗ = ( - î + 3k̂ ) + λ ( î - aĵ ) and r⃗ = (… Let P be the point (10,-2,-1) and Q be the foot of the perpendicular drawn from the… The distance of the point (-1,2,3) from the plane r⃗ ·(î-2 ĵ+3 k̂)=10 parallel to the… Let O be the origin, and M and N be the points on the lines (x-5)/4=(y-4)/1=(z-5)/3 and… If the shortest distance between the lines (x+2)/2=(y+3)/3=(z-5)/4 and… Let the values of λ for which the shortest distance between the lines (x-1)/2 = (y-2)/3… If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4)… One vertex of a rectangular parallelopiped is at the origin O and the lengths of its… The line of shortest distance between the lines (x-2)/0=(y-1)/1=z/1 and… The shortest distance between the lines (x-4)/4=(y+2)/5=(z+3)/3 and… Let the shortest distance between the lines L: (x-5)/-2=(y-λ)/0=(z+λ)/1, λ ≥ 0 and L₁… The shortest distance between the lines (x+7)/-6=(y-6)/7=z and (7-x)/2=y-2=z-6 is : The perpendicular distance, of the line (x-1)/2=(y+2)/-1=(z+3)/2 from the point… The distance of the line (x-2)/2=(y-6)/3=(z-3)/4 from the point (1,4,0) along the line… The sum of all values of α, for which the shortest distance between the lines… The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the… The shortest distance between the lines r⃗=(1/3î+2ĵ+8/3k̂)+λ(2î-5ĵ+6k̂) and… If the shortest distance between the lines (x-λ)/3=(y-2)/-1=(z-1)/1 and… Let P(α,β,γ) be the point on the line (x-1)/2=(y+1)/-3=z at a distance 4√14 from the… Angle Between Planes Distance of Point from Plane If the mirror image of the point (1, 3, 5) with respect to the plane 4x - 5y + 2z = 8 is… A plane E is perpendicular to the two planes 2 x-2 y+z=0 and x-y+2 z=4, and passes… The equation of the planes parallel to the plane x - 2y + 2z - 3 = 0 which are at unit… Let the image of the point P(2,-1,3) in the plane x+2 y-z=0 be Q. Then the distance of… Let Q be the foot of perpendicular drawn from the point P(1,2,3) to the plane x+2… If the lines r⃗ = ( î - ĵ + k̂ ) + λ ( 3ĵ - k̂ ) and r⃗ = ( α î - ĵ ) + μ ( 2î - 3k̂ )… Let N be the foot of perpendicular from the point P(1,-2,3) on the line passing through… Let (x - 2)/3 = (y + 1)/(- 2) = (z + 3)/(- 1) lie on the plane px - qy + z = 5, for some… Let d be the distance between the foot of perpendiculars of the points P(1, 2, -1) and… If the mirror image of the point (2, 4, 7) in the plane 3x - y + 4z = 2 is (a, b, c)… Let the foot of perpendicular from the point A(4,3,1) on the plane P: x-y+2 z+3=0 be N… If for a > 0, the feet of perpendiculars from the points A(a, -2a, 3) and B(0, 4, 5) on… Let P be the plane, passing through the point (1,-1,-5) and perpendicular to the line… Let P be the plane passing through the points (5,3,0),(13,3,-2) and (1,6,2). For α ∈ ℕ… Let P be the plane containing the straight line (x-3)/9=(y+4)/-1=(z-7)/-5 and… Let a plane P pass through the point (3, 7, -7) and contain the line, (x - 2)/(- 3) = (y… If the equation of a plane P, passing through the intersection of the planes, x + 4y - z… The foot of the perpendicular from a point on the circle x²+y²=1, z=0 to the plane 2 x+3… The distance of the point (7, -3, -4) from the plane passing through the points (2, -3… Let the line x/1=(6-y)/2=(z+8)/5 intersect the lines (x-5)/4=(y-7)/3=(z+2)/1 and… More Mathematics chapters Frequently asked questions How many Three Dimensional Geometry questions are there for JEE Main? PrepWiser has 297 Three Dimensional Geometry questions for JEE Main, grouped into 7 topics such as Direction Cosines and Ratios, Equation of Line in 3D and Equation of Plane.
Are these Three Dimensional Geometry questions from previous-year JEE Main papers? Yes — most are genuine previous-year questions from past JEE Main papers, and the rest are practice questions written to match the JEE Main pattern and difficulty.
Where can I see the answers and solutions to these Three Dimensional Geometry questions? Open any Three Dimensional Geometry question in PrepWiser (it is free) to attempt it; most come with a step-by-step solution.