Three Dimensional Geometry — practice questions

157 curated practice questions from Three Dimensional Geometry (Mathematics JEE Main, NTA Exams) — 132 easy, 15 medium, 10 hard. Every question includes the final answer free; step-by-step worked solutions with a free account.

• If position vectors of A, B, C, D are respectively 2 i +3 j +5 k , i +2 j +3 k ,-5 i +4 j -2 k and i +10 j +10 k , then
• Let a , b and c be three unit vectors such that a ( b c )= 3 2 ( b + c ) . If b is not parallel to c , then the angle…
• If θ is an angle given by = ^ 2 z+ ^ 2 + ^ 2 ^ 2 z+ ^ 2 + ^ 2 , where x, , Y are the angles made by a line with the…
• If the coordinates of the points A, B, C be (–1, 3, 2), (2, 3, 5) and (3, 5, –2) respectively, then ∠A =
• Let a , ~b and c be three non-zero vectors such that no two of them are collinear and ( a b ) c = 1 3 | ~b || c | a …
• The point of intersection of the line x 1 = y-1 2 = z+2 3 and the plane, 2x + 3y + z = 0, is :
• If a =3 i -2 j + k and b =- i + j + k , then the unit vector parallel to a + b , is
• If the vector b is collinear with the vector a =(2 2 ,-1,4) | b |=10 , then
• The direction ratios of two parallel lines are 4,-3,-1 , & + _ 1 1+ _ 1 2 . The value of pair ( , ) is
• If a line makes angles x, , Y with the co-ordinate axes, then 2 + 2 + 2 is
• Equation of the line passing through (1,1,1) and perpendicular to 2 x+3 y+z+5=0 is :
• If | a b |= a ~b then angle between a and b is
• A line makes acute angles of α, β and γ with the co-ordinate axes such that cos α cos β = cos β cos = 2 9 and = 4 9 …
• The angle between the lines x-z z = y+1 -z , z=z and x-1 1 = 2 y+3 2 = 2+5 2 is
• The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to
• If the angle between the line 2(x+1) = y = z + 4 and the plane is 6 , then the value of λ is
• If ABCDE is a pentagon, then AB + AE + BC + DC + ED + AC equals
• The unit vector perpendicular to vectors i + j and j + k is
• If line r =( i -2 j - k )+ (2 i + j +2 k ) is parallel to the plane r (3 i -2 j - m k )=4 then the value of m is
• If a and b are vectors of equal magnitude 2 and α be the angle between them, then magnitude of a + b will be 2 if
• If the vectors a , ~b and c form the sides BC, CA and AB respectively of a triangle ABC, then
• Consider points A, B, C and D with position vectors 7 i -4 j +7 k , i -6 j +10 k , - i -3 j +4 k and 5 i - j +5 k…
• The equation of the plane through the intersection of the planes x + 2y + 3z = 4 and 2x + y – z = –5 and perpendicular…
• The coordinates of the points A, B, C, D are (4, α, 2), (5, –3, 2), (β, 1, 1) & (3, 3, –1). Line AB would be…
• A line makes angles α,β,γ with the coordinate axes. If α + β = 90º, then γ =
• A symmetrical form of the line of intersection of the planes x = ay + b and z = cy + d is:
• The angle between the vectors a + b & a - b , given | a |=2,| ~b |=1 and angle between a ~b is π/3, is
• The plane 2x – 3y + 6z – 11 = 0 makes an angle sin –1 (α) with x-axis. The value of α is equal to
• If θ be the angle between vectors i +2 j +3 k and 3 i +2 j + k , then the value of sin θ is
• The direction cosines l_ 4 m_ 4 n of two lines are connected by the relations 1+m+n=0 , m=0 , then the angle between…
• A line makes equal angles with co-ordinate axes. Direction cosines of this line are
• Let a = j - k and c = i - j - k . Then the vector b satisfying a b + c =0 and a b =3, is
• If the angle between a and b is θ then for a b 0
• If the lines x+2 4 +1 = y-1 4 = z -18 and x -3 = y+1 5 -3 = z-1 6 are parallel to each other then the value of the pair…
• The angle between the planes, 2x – y + z = 6 and x + y + 2z = 7 is
• The direction ratios of two perpendicular lines are 1,-3,5 and _ 1 1+ _ 1 2+ . Then is
• The equation of the plane containing the two lines x-1 2 = y+1 -1 = z-0 3 and x -2 = y-2 11 = z+1 -1 is
• The distance between the parallel planes a x+b y+c z+d=0 and a x+b y+c z+d^ =0 is :
• If A= [p, q, r^ ] and B= [p^ , q^ , r^ ] are two points on the line x= y=y z , such that O A=3 , O B=4 , then p p^ +q…
• The point at which the line joining the points (2, –3, 1) & (3, –4, –5) intersects the plane 2x + y + z = 7 is :
• A line AB in three-dimensional space makes angles 45 o and 120 o with the positive x - axis and the positive y^ - axis…
• A line passes through the points (6,-7,-1) and (2,-3,1) . The direction cosines of the line, so directed that the angle…
• The projections of a vector on the three coordinate axes are 6,-3,2 respectively. The direction cosines of the vector…
• If a and b are two vectors, then -
• The angle between the lines joining the points (1,0,-3),(2,-1,2) and (1,1,1),(3,2,0) respectively, is
• The point A(3, -2, 4) is shifted parallel to the line x 3 = y-1 2 = z+1 3 by a distance 1 to the point P. The…
• A line passes through a point A with p.v. 3 i + j - k and is parallel to the vector 2 i - j +2 k . If P is a point on…
• |( A + B )|^ 2 +|( A - B )|^ 2 equals
• ABC is a triangle where A=(2,3,5), B=(-1,3,2) and C=( , 5, ) . If the median through A is equally inclined with the…
• Projection vector of a on b is
• Given the vectors a ~b the angle between which equals 120° . If | a |=3 | b |=4 then the length of the vector 2 a - 3 2…
• If (2,3,5) is one end of a diameter of the sphere x^ 2 +y^ 2 +z^ 2 -6 x-12 y-2 z+30=0 , then the coordinates of the…
• For any two vector a and b , correct statement is
• The direction cosines of a line whose equation is x-1 2 = y+3 4 = z-2 -3 , are
• The vector equation of the plane passing through the origin and the line of intersection of the plane r a = and r b =…
• If a line makes an angle of 4 with the positive directions of each of x - axis and y^ - axis, then the angle that the…
• If a × b = a × c then correct statement is
• The points A(1,2,-1), B(2,5,-2), C(4,4,-3) and (3,1,-2) are
• If a b = c d and a c = b d and a d , ~b c , Then
• The angle between the straight lines whose direction cosines are given by 2|+2 m-n=0, m n+n|+ m=0, is
• If a plane passes through the point (1, 1, 1) and is perpendicular to the line x-1 3 = y-1 0 = z-1 4 then its…
• If the direction ratios of a line are 1,-3,2 , then the direction cosines of the line are :
• The distance of point of intersection of lines x-4 1 = y+3 -4 = z+1 7 and x-1 2 = y+1 -3 = z+10 8 from the point…
• If ^ , ^ , y y^ are not all equal, the point of intersection of the lines x-x^ = y- ^ = z-y^ y and x- x^ = y- ^ = z-y…
• Find the direction cosines of the line x-2 2 = 2 y-5 -3 , z=-1
• The volume of the parallelopiped whose sides are given by a =2 i -3 j . B = i + j - k and O =3 i - k is :
• The vector a = i +2 j + k lies in the plane of the vectors b = i + j and c = j + k and bisects the angle between and c…
• The line, x-2 3 = y-3 4 = z-4 5 is parallel to the plane :
• If | a b | = 3 and | a b | = 4, then the angle between a and is
• | a | a |^ 2 - b | ~b |^ 2 |^ 2 =
• Let P be the plane, which contains the line of intersection of the planes, x+y+z-6=0 and 2 x+3 y+z+5=0 and it is…
• The Plane 2x – (1 + λ) y + 3λ z = 0 passes through the intersection of the planes
• The direction ratio of normal to the plane through (1, 0, 0), (0, 1, 0), which makes an angle 4 with plane x + y = 3…
• The image of the points (–1, 3, 4) in the plane x – 2y = 0 is
• Image of the point P with position vector 7 i - j +2 k in the line whose vector equation is, r =9 i +5 j +5 k + ( i +3…
• If a =4 i -2 j +3 k and b =-8 i +4 j -6 k are two vectors then a , b are
• The equations of lines passing through the points (-2,1,0),(3,4,-1) and -7,-2,1
• The length of the perpendicular from origin on plane r (3 i -4 j +12 k )=5 is
• If i + j + k , 2 i +5 j , 3 i +2 j -3 k and i -6 j - k be p.v. of four points A,B,C and D respectively, then the angle…
• The shortest distance between the skew lines _ 1 : r = a _ 1 + b _ 1 and _ 2 : r = a _ 2 + b _ 2 is:
• The equation of a plane which passes through (2, –3, 1) and is normal to the line joining the points (3, 4, –1) and (2…
• The shortest distance between the plane 12 x+4 y+3 z=327 and the sphere x^ 2 +y^ 2 +z^ 2 +4 x-7 y-6 z=155 is
• Two lines r = a _ 1 + b _ 1 and r = a _ 2 + b _ 2 would be coplanar if:
• If A=(5,-1,1), B=(7,-4,7), C=(1,-6,10), =(-1,-3,4) then ABCD is a
• A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z. The…
• If the vectors 3 i + j -2 k , i +2 j -3 k , 3 i + j +5 k are co-planar, then the value of λ is :
• If the given planes, ax + by + cz + d = 0 and a´x + b´y + c´z + d´ = 0 be mutually perpendicular, then:
• The points with position vectors 3 i -4 j -4 k , 2 i - j + k and i -3 j -5 k form
• The direction ratios of two lines are a_ 1 b_ 1 c and 1 bc , 1 ca , 1 ab respectively. Then lines are
• If a point R(4, y, z) lies on the line segment joining the points P(2, -3, 4) and Q(8, 0, 10), then the distance of R…
• The angle between the line, x+1 3 = y-1 2 = z-2 4 and the plane, 2x + y – 3z + 4 = 0, is :
• A line makes the same angle , with each of the x and 2 axes. If the angle , which it makes with Y -axis, is such that ^…
• The direction cosines of the line joining the points (2,3,-1) & (3,-2,1) are
• The value of k, if the line x-4 1 = y-z 1 = z-k 1 lies on the plane 2 x-4 y+z=7 , will be
• Direction cosines of normal to the plane containing lines x=y=z and x -1= y -1= z -1 d (where d E R- 1 , are
• Equation of the plane passing through the line of intersection of the planes P : ax + by + cz + d = 0, P’ : a’ x + b’ y…
• The distance of the point (2,0,-3) from the plane 5 x-12 y=0 is
• If the straight lines x = 1 + s, y = −3 − λ s, z = 1 + λs and x = t 2 , y = 1 + t, z = 2 – t, with parameters s and t…
• If, D, E, F are mid points of sides BC, CA and AB respectively of a triangle ABC, and i + j , j + k , k + i are p.v. of…
• For non-zero vectors a , b , c ,|( a b ) c |=| a || b || c | holds if and only if
• If for vectors a and b , a × b = 0 and . b = 0, then
• The equation of the plane passing through the points (3, 2, 2) and (1, 0 –1) and parallel to the line x-1 2 = y-1 -2 =…
• If a line makes angles x, , Y with the positive direction of x, y and z-axes respectively. Then ^ 2 x+ ^ 2 + ^ 2 is…
• The direction cosines of a line equally inclined with co-ordinate axes are :
• Two system of rectangular axes have the same origin. If a plane cuts them at distances a_ 1 b_ 1 c and a^ , b^ , c^…
• If a =2 i +5 j and b =2 i - j , then unit vector in the direction of a + b is
• A plane meets the coordinate axes in A, B, C and (α,β,γ) is the centroid of the triangle ABC. Then the equation of the…
• The plane ax + by + cz = 1 meets the co-ordinate axes in A, B and C. The centroid of the triangle is :
• If A = (p, q, r) and B = (p^ , q^ , r^ ) are two points on the line x=t y=v z such that OA = a , OB = b then p p^ +q q^…
• Given below are two statements. STATEMENT - 1: If 'a' is an integer, then the straight lines r = i +2 j +3 k + (a i +2…
• Unit vector perpendicular to the plane of the triangle ABC with pv’s a , ~b , c of the vertices A, B, C is
• The point of intersection of lines, x -4 5 = y -1 2 = z 1 x -1 2 = y -2 3 = z -3 4 is :
• The distance of the point (1, –5, 9) from the plane x – y + z = 5 measured along a straight line x = y = z is
• If ( a b ) c = a ( b c ) , where a , b , c are any three vectors such that ≠ 0, b c ≠ 0, then a and c are
• The equation of the plane through the intersection of the planes ax + by + cz + d = 0 and l x + my + nz + p = 0 and…
• Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
• If the points A(1,2,-1) , E (2,6,2) and [( ,-2,-4) are collinear, then is
• The length of the perpendicular from the point (2, -1, 4) on the straight line, x+3 10 = y-2 -7 = z 1 is:
• A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(-1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR…
• The vector c , directed along the internal bisector of the angle between the vectors, a =7 i -4 j -4 k and b =-2 i - j…
• The vector equation of the plane is r [2 i -3 j +6 k ]=13 ; where 2 i -3 j +6 k is normal to the plane. Find the length…
• Equation of the plane which passes through the point of intersection of lines x-1 3 = y-2 1 = z-3 2 and x-3 1 = y-1 2 =…
• The equation of the plane containing the line x - x _ 1 = y - y _ 1 ~m = z - z _ 1 n is a (x – x 1 ) + b(y – y 1 ) +…
• If i , j , k are position vectors of A, B, C and AB = CX , then position vector of X is
• The position vector of the vertices of triangle ABC are i , j and k then the position vector of its orthocentre is
• If a line has direction ratios > , then its direction- cosines will be :
• If | a |=| b |, then is
• The distance of the point, (–1, –5, –10) from the point of intersection of the line, x-2 3 = y+1 4 = z-2 12 and the…
• If a , b , c are any vectors then which one of the following is a wrong statement.
• The equation of the plane which is right bisector of the line joining (2, 3, 4) and (6, 7, 8), is :
• The distance between the line r =2 i -2 j +3 k + [ i - j +4 k ] and the plane r [ i +5 j + k ]=5 i 5
• Find the angle between the two straight lines, r =3 i -2 j +4 k + (-2 i + j +2 k ) and r = i +3 j -2 k + (3 i -2 j +6 k…
• The lines, r _ 1 = i +2 j +3 k + (2 i +3 j +4 k ) and I _ 2 =2 i +3 j +4 k + (3 i +4 j +5 k ) are :
• The angle between the line x-2 a = y-2 b = z-2 c and the plane ax + by + cz + 6 = 0 is
• Two non zero vectors a and b will be parallel, if
• If a = i +3 j -2 k and b =4 i -2 j +4 k , then (2 a + b ) . ( a – 2 b ) equals
• If the position vector of points A and B with respect to point P are respectively a and b then the position vector of…
• If the lines, x-1 2 = y+1 3 = z-2 4 meets the plane, x+2 y+3 z=15 at a point P, then the distance of P from the origin…
• The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines r =( i + j )+ i +2 j …
• If | array lll a & a^ 2 & 1+a^ 3 b & b^ 2 & 1+b^ 3 c & c^ 2 & 1+c^ 3 array |=0 and vectors (1, a, a^ 2 ), (1, b, b^ 2 )…
• The angle between the lines 2 x=3 y=-z and 6 x=-y=-4 z is
• A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the co-ordinate…
• The angle between the lines x+z 1 = y+3 -z = z-4 1 and x-1 -1 = y 1 = z 0 is
• The position vector of two points P and Q are respectively p and q then the position vector of the point dividing PQ in…
• If the distance between planes, 4x - 2y - 4z + 1 = 0 and 4x - 2y - 4z + d = 0 is 7, then d is:
• Vectors a ~b make an angle = 2 3 . If | a |=1,| ~b |=2 then ( a +3 b ) (3 a - b )^ 2 =
• Two lines, whose direction ratios are : 1 , b 1 , c 1 > and 2 , b 2 , c 2 > respectively are perpendicular if
• Distance between two parallel planes 2 x+y+2 z=8 and 4 x+2 y+4 z+5=0 is
• Find the equation of the line which passes through (1,2,3) and is parallel to the line whose equation is x=a y+b , z=c…
• The angle between two lines x z = y z = z -1 and x-1 z = y-1 1 = z-1 z is
• The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is
• In a regular hexagon ABCDEF, AB = a , BC = b and CD = c . Then, AE =
• If the lines x-2 1 = y-3 1 = z-4 -k and x-1 k = y-4 2 = z-5 1 are coplanar, then k can have
• The image of the point P (1, 3, 4) in the plane 2x – y + z + 3 = 0 is
• Given below are two statements. STATEMENT - 1: The direction cosines of one of the angular bisectors of two…
• For any vectors a , ; | a × b | 2 + ( a . b ) 2 is equal to
• Angle between the vectors 2 i +6 j +3 k and 12 i -4 j +3 k is

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