Three Dimensional Geometry — practice questions

91 curated practice questions from Three Dimensional Geometry (Mathematics JEE Advanced, NTA Exams) — 68 hard, 23 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• If P 1 : r n _ 1 - d 1 = 0, P 2 : r n _ 2 – d 2 = 0 and P 3 : r n _ 3 – d 3 = 0 are three planes and n _ 1 , n _ 2 and…
• The equation of a plane passing through the line of intersection of the planes x + 2 y + 3 z = 2 and x – y + z = 3 and…
• Let L 1 and L 2 denote the lines r = i + (- i +2 j +2 k ), R and r = (2 i - j +2 k ), R respectively. If L 3 is a line…
• Let a =a_ 1 i +a_ 2 j +a_ 3 k , b = b _ 1 i + b _ 2 j + b _ 3 k and c = c _ 1 i + c _ 2 j + c _ 3 k be three non–zero…
• If a ~b lie on a plane normal to the plane containing c ~d then, ( a b ) ( c d ) is equal to
• Three vectors a , b and c are such that a b = c , b c = a , c a = b . Answer the following questions : Vectors 2 a -3 b…
• Consider a pyramid OPQRS located in the first octant ( x ≥ 0, y ≥ 0, z ≥ 0) with O as origin, and OP and OR along the x…
• A line with positive direction cosines passes through the point P (2, –1, 2) and makes equal angles with the coordinate…
• If a , ~b and c are any three vectors, then a ( b c )=( a b ) c is true if :
• Assertion : Let a , ~b , c and d are position vectors of four points A, B, C and D and 3 a -2 ~b +5 c -6 ~d = 0 , then…
• Let P 1 : 2 x + y − z = 3 and P 2 : x + 2 y + z = 2 be two planes. Then, which of the following statement(s) is (are)…
• If a vector a is expressed as the sum of two vectors a ^ and a ^ along and perpendicular to a given vector b then a ^ is
• Equation of 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 = z-2…
• Equation of line in the plane P : 2x – y + z – 4 = 0 which is perpendicular to the line l whose equation is x-2 1 = y-2…
• If the vector product of a constant vector A with a variable vector in a fixed plane OAB be a constant vector, then…
• Two given points P and Q in the rectangular cartesian co–ordinates lie on y = 2 x + 2 such that O P i =-1 and O Q i =+2…
• 338. A straight line drawn from the point P (1, 3, 2), parallel to the line x-2 1 = y-4 2 = z-6 1 , intersects the…
• In R 3 , let L be a straight line passing through the origin. Suppose that all the points on L are at a constant…
• Let α, β, γ, δ be real numbers such that α 2 + β 2 + γ 2 ≠ 0 and α + γ =1. Suppose the point (3, 2, −1) is the mirror…
• If a and b unequal unit vectors such that ( a - b ) [ b + a ) (2 a + b ]= a + b , then angle θ between and b is
• If the unit vectors e _ 1 and e _ 2 are inclined at an angle 2θ and | e _ 1 - e _ 2 |<1 , then for θ ∈ [0, π], θ may…
• The point P is the intersection of the straight line joining the points Q (2, 3, 5) and R (1, –1, 4) with the plane 5 x…
• Let a 1 x + b 1 y + c 1 z + d 1 = 0 and a 2 x + b 2 y + c 2 z + d 2 = 0 be two planes, where d 1 , d 2 > 0 . Then…
• If a plane cuts off intercepts OA = a, OB = b, OC = c from the coordinate axes, then the area of the triangle ABC =
• If the distance of the point P (1, –2, 1) from the plane x + 2 y – 2 z = α , where α > 0, is 5, then the foot of the…
• Assertion : A point on the straight line 2x + 3y – 4z = 5, 3x – 2y + 4z = 7 can be determined by taking x = k and then…
• Assertion : Let a =3 i - j , b =2 i + j -3 k . If b = b _ 1 + b _ 2 such that b _ 1 is collinear with a and b _ 2 is…
• The direction ratio’s of the line x – y + z – 5 = 0 = x – 3y – 6 are
• Let be vectors of length 3, 4 and 5 respectively. Let be perpendicular to B + C , ~B to C + A and C to A + B . Then…
• Let P (3, 2, 6) be a point in space and Q be a point on the line r =( i - j +2 k )+ (-3 i + j +5 k ) . Then the value…
• If a , ~b , c be the unit vectors such that b is not parallel to c and a (2 ~b c )= b then the angle that a makes with…
• Let p is the p.v. of the orthocentre & g is the p.v. of the centroid of the triangle ABC where circumcentre is the…
• Let A ( a ) and B ( b ) be points on two skew lines r = a + p and r = b + u q and the shortest distance between the…
• [( a b ) ( b c )( b c ) ( c a )( c a ) ( a b )]=
• Let a = i + j and b =2 i - k . The point of intersection of the lines r a = b a and r b = a b is :
• If a , ~b , c and d are the pv’s of the point A, B, C and D respectively in three dimensional space and satisfy the…
• Consider a tetrahedron with faces f 1 , f 2 , f 3 , f 4 . Let a _ 1 , a _ 2 , a _ 3 , a _ 4 be the vectors whose…
• Assertion : If a =3 i + k , ~b =- i +2 j + k , c = i + j + k and d =2 i - j , then there exist real numbers α, β, γ…
• The length of the projection of the line segment joining the points (5, –1, 4) and (4, –1, 3) on the plane x + y + z =…
• a and b are two non collinear unit vectors. Then a , b , x a -y b form a triangle, if:
• If line makes angle α,β,γ,δ with the four diagonals of a cube, then the value of cos 2 α + cos 2 β + cos 2 γ + cos 2 δ =
• A vector ( d ) is equally inclined to three vectors a = i - j + k , b =2 i + j and c =3 j -2 k . Let x , y , z be three…
• If a , ~b and c are three unit vectors equally inclined to each other at an angle α. Then the angle between a and plane…
• If the vectors a i + j + k , i +b j + k and i + j +c k (a ≠ b ≠ c ≠ 1) are coplanar then the value of 1 1-a + 1 1-b + 1…
• P is fixed point (a, a, a) on a line through the origin equally inclined to the axes, then any plane through P…
• If the sum of the squares of the distance of a point from the three coordinate axes be 36, then its distance from the…
• If a b are unit vectors such that ( a + b ) (2 a +3 b ) (3 a -2 b )=0 then angle between a and b
• A plane which is perpendicular to two planes 2 x – 2 y + z = 0 and x – y + 2 z = 4, passes through (1, –2, 1). The…
• ( r i )( i r )+( r j )( j r )+( r k )( k r )=
• Three lines L_ 1 : r = i , R, L_ 2 : r = k + j , R and L_ 3 : r = i + j +v k , v R are given. For which point(s) Q on L…
• For a non zero vector A if the equations A B = A C and A B = A C hold simultaneously, then
• If the lines x-1 2 = y+1 3 = z-1 4 and x-3 1 = y-k 2 = z 1 intersect, then the value of k is
• If b c are any two perpendicular unit vectors and a is any vector, then, ( a b ) b +( a c ) c + a ( b c ) | b c |^ 2 (…
• Let a 1 x + b 1 y + c 1 z + d 1 = 0 and a 2 x + b 2 y + c 2 z + d 2 = 0 be two planes, where d 1 , d 2 > 0 . Then…
• If a +2 b +3 c = o , then a b + b c + c a is equal to
• A, B, C and D are four points in a plane with pv ^ s a , ~b , c and d respectively such that ( a - d ) ( b - c )=( b …
• The value of k such that x-4 1 = y-2 1 = z-k 2 lies in the plane 2 x – 4 y + z = 7, is
• The vectors p q satisfy the system of equations 2 p + q = a , p +2 q = b and the angle between p q is θ. If it is known…
• ‘P’ is a point inside the triangle ABC such that BC ( PA )+ CA ( ~PB )+ AB ( PC )=0 then for the triangle ABC the point…
• Let P be the image of the point (3, 1, 7) with respect to the plane x – y + z = 3. Then the equation of the plane…
• Three vectors a , b and c are such that a b = c , b c = a , c a = b . Answer the following questions : Let x = a + b …
• Let S be the reflection of a point Q with respect to the plane given by r =-(t+p) i +t j +(1+p) k , where t and p are…
• Let A(1, 1, 1), B(2, 3, 5), C(–1, 0, 2) be three points, then equation of a plane parallel to the plane ABC which is at…
• The straight lines x-1 1 = y-2 2 = z-3 3 and x-1 2 = y-2 2 = z-3 -2 are
• If z _ 1 =a i +b j & z _ 2 =c i +d j are two vectors in i j system where | z _ 1 |= | z _ 2 |= r & z _ 1 z _ 2 =0 then…
• Let r , a , ~b and c be four non–zero vectors such that r a =0, | r b |=| r || b |,| r c |=| r || c |, then [a b c] =
• Let a , ~b and c be non–coplanar unit vectors equally inclined to one another at an acute angle θ. Then [ array lll a &…
• Three vectors a , b and c are such that a b = c , b c = a , c a = b . Answer the following questions : a ( b c )+ b ( c…
• If (( a b ) ( c d )) ( a d )=0, then which of the following is always true
• For any vector A , i ( i A )+ j ( j A )+ k ( k A ) simplifies to
• Perpendiculars are drawn from points on the line x+2 2 = y+1 -1 = z 3 to the plane x + y + z = 3. The feet of…
• If a _ 1 , a _ 2 and a _ 3 are non–coplanar vectors and (x + y – 3) a _ 1 + (2x – y + 2) a _ 2 + (2x + y + λ) a _ 3 = 0…
• Three vectors a , b and c are such that a b = c , b c = a , c a = b . Answer the following questions : If vector 3 a -2…
• is prependicular to DB then ( BD AC ) ( OD OC ) is :
• The vectors a , ~b , c are of the same length & pairwise form equal angles. If a = i + j & b = j + k then c can be
• a b are two given unit vectors at right angle. The unit vector equally inclined with a b and a b will be :
• Let a 1 x + b 1 y + c 1 z + d 1 = 0 and a 2 x + b 2 y + c 2 z + d 2 = 0 be two planes, where d 1 , d 2 > 0 . Then…
• The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2 x + y – 2 z = 5 and 3 x…
• The value of a so that the volume of the parallelopiped formed by the vectors i +a j - k , j +a k , i + j + k becomes…
• If A 1 , A 2 , A 3 , ......, A n are the vertices of a regular plane polygon with n sides & O is its centre then _ i=1…
• ( u v ) ( u v ) is equal to
• Assertion : Let a =2 i +3 j - k , b =4 i +6 j -2 k , then a b = o Reason : If a o , b o and a and b are non–collinear…
• In R 3 , consider the planes P 1 : y = 0 and P 2 : x + z = 1. Let P 3 be a plane, different from P 1 and P 2 , which…
• Let L 1 be the line of intersection of the planes given by the equations 2 x +3 y + z = 4 and x +2 y + z = 5 . Let L 2…
• The equation of the plane which bisects the angle between the planes 3x – 6y + 2z + 5 = 0 and 4x – 12y + 3z – 3 = 0…
• A variable plane passes through a fixed point (1, 2, 3). The locus of the foot of the perpendicular drawn from origin…
• The triple product ( d + a ) [ a ( b ( c d ))] simplifies to
• Let a = i - k , b =x i + j +(1-x) k and c =y i +x j +(1+x-y) k . Then [ array lll a & ~b & c array ] depends on
• If a , ~b , c are such that [ array lll a & b & c array ]=1, c = a b , a ~b < 2 3 and | a |= 2 ,| ~b |= 3 , | c |= 1 3…
• If the foot of the perpendicular from the origin to a plane is P (a, b, c), the equation of the plane is
• Equation of the plane containing the straight line x 2 = y 3 = z 4 and perpendicular to the plane containing the…

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