MCQ Questions Chapter 11 Three Dimensional Geometry Class 12 Mathematics

Please refer to MCQ Questions Chapter 12 Three Dimensional Geometry Class 11 Mathematics with answers provided below. These multiple-choice questions have been developed based on the latest NCERT book for class 11 Mathematics issued for the current academic year. We have provided MCQ Questions for Class 11 Mathematics for all chapters on our website. Students should learn the objective based questions for Chapter 12 Three Dimensional Geometry in Class 11 Mathematics provided below to get more marks in exams.

Chapter 12 Three Dimensional Geometry MCQ Questions

Please refer to the following Chapter 12 Three Dimensional Geometry MCQ Questions Class 11 Mathematics with solutions for all important topics in the chapter.

Question. The shortest distance between the lines r = (i + j− k) +λ (3i − j) and r = (4i − k) +μ (2i + 3k) is:
(a) 6
(b) 0
(c) 2
(d) 4

Question. The plane which passes through the point (3,2,0) and the line x–3/1 = y–6/5 = z-4/4 is:
(a) x − y + z =1
(b) x + y + z = 5
(c) x + 2y − z =1
(d) 2x − y + z = 5

Question. The intersection of the spheres x² + y² + z² + 7x −2y − z =13 and x² + y² + z² − 3x + 3y + 4z = 8 is the same as the intersection of one of the sphere and the plane:
(a) 2x − y − z =1
(b) x − 2y − z =1
(c) x − y − 2z =1
(d) x − y − z =1

Question. 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 coordinates of each of the points of intersection are given by:
(a) (2a, 3a, 3a) (2a, a, a)
(b) (3a, 2a, 3a) (a, a, a)
(c) (3a, 2a, 3a) (a, a, 2a)
(d) (3a, 3a, 3a) (a, a, a)

Question. If the line x–1/2 = y+1/3 = z-1/4 , x-3/1 = y-k/2 = z/1 intersect, then k = ?
(a) 2/9
(b) 9/2
(c) 0
(d) –1

Question. The co-ordinates of the foot of the perpendicular drawn from the point A(1, 0, 3) to the join of the points B(4, 7, 1) and C(3, 5, 3) are:
(a) (5/3, 7/3, 17/3)
(b) (5, 7, 17)
(c) (5/3, –7/3, 17/3)
(d) (–5/3, 7/3, –17/3)

Question. The equation of the plane passing through the lines x–4/1 = y–3/1 = z-2/2 and x–3/1 = y–2/–4 = z/5 is:
(a) 11x − y − 3z = 35
(b) 11x + y − 3z = 35
(c) 11x − y + 3z = 35
(d) None of these

Question. The line x–2/3 = y–3/4 = z-4/5 is parallel to the plane:
(a) 3x + 4y + 5z = 7
(b) 2x + y − 2z = 0
(c) x + y − z = 2
(d) 2x + 3y + 4 z = 0

Question. If P = (0, 1, 0), Q =(0, 0, 1), then projection of PQ on the plane x + y + z = 3 is:
(a) √3
(b) 3
(c) √2
(d) 2

Question. The length of the perpendicular from the origin to line r = (4i + 2j+ 4k) +λ (3i + 4j−5k) is:
(a) 2√5
(b) 2
(c) 5√2
(d) 6

Question. The shortest distance between the lines x–1/2 = y–2/3 = z-3/4 , x-2/3 = y-4/4 = z–5/5 is :
(a) 1/6
(b) 1/√6
(c) 1/√3
(d) 1/3

Question. If the straight lines x =1+ s, y = 3−λs, z =1+λs and x = t/2 ,y =1+ t, z = 2 −t with parameters s and t respectively, are co-planar, then λ equals:
(a) 0
(b)–1
(c) –1/2
(d) –2

Question. The ratio in which the plane x − 2y + 3z = 17 divides the line joining the point (–2, 4, 7) and (3, –5, 8) is:
(a) 10 : 3
(b) 3 : 1
(c) 3 : 10
(d) 10 : 1

Question. The xy-plane divides the line joining the points (–1, 3, 4) and (2, –5, 6)
(a) Internally in the ratio 2:3
(b) Internally in the ratio 3:2
(c) Externally in the ratio 2:3
(d) Externally in the ratio 3:2

Question. The distance between the line r = (i + j + 2k) +λ (2i + 5j + 3k) and the plane r.(2i + j − 3k) = 5 is:
(a) 5/√14
(b) 6/√14
(c) 7/√14
(d) 8/√14

Question. The equation of the plane, which makes with co-ordinate axes a triangle with its centroid (α, β, γ), is:
(a) α x +β y +γ z = 3
(b) x/α +y/β +z/γ = 3
(c) α x +β y +γ z =1
(d)  x/α +y/β +z/γ = 3

Question. Angle between two planes x +2y+2z=3 and −5x + 3y + 4z = 9 is:
(a) cos^–1 3√2/10
(b) cos^–1 19√2/30
(c) cos^–1 9√2/20
(d) cos^–1 3√2/5

Question. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is:
(a) 9/2
(b) 5/2
(c) 7/2
(d) 3/2

Question. A tetrahedron has vertices at O(0,0, 0), A(1,2,1), B(2, 1, 3) and C(–1, 1, 2). Then the angle between the faces OAB and ABC will be:
(a) cos−1(19/35)
(b) cos−1(17/31)
(c) 30°
(d) 90°

Question. The distance of the point (2, 1, –1) from the plane x − 2y + 4z = 9 is:
(a) √13/21
(b) 13/21
(c) 13/√21
(d) √(13/21)

Question. The cartesian equations of a line are 6x − 2 . = 3y +1 = 2z − 2 The vector equation of the line is:

(b) r = (3i −3j+ k) +λ (i + 2j+ 3k)
(c) r = (i + j+ k) +λ (i + 2j+ 3k)
(d) None of these

Question. The sine of angle between the straight line x–2/3 = y–3/4 = z-4/5 and the plane 2x − 2 y + z = 5 is:
(a) 2/√3/5
(b) √2/10
(c) 4/5√2
(d) 10/6√5

Question. A unit vector perpendicular to plane determined by the points P(1, –1, 2), Q(2, 0, –1) and R(0, 2, 1) is:

Question. The reflection of the point (2, –1, 3) in the plane 3x − 2y − z = 9 is: 

Question. A non-zero vector a is parallel to the line of intersection of the plane determined by the vectors i, i + j and the plane determined by the vectors i – j, i + k. The angle between a and the vector i – 2j + 2k is:
(a) π/4 or 3π/4
(b) 2π/4 or 3π/4
(c) π/2 or 3π/2
(d) None of these

Question. Two lines L1 : x=5, y/3−α = z/−2 and L2 : x=α, y/−1 = z/2−α are coplanar. Then, α can take value(s):     
(a) 1
(b) 2
(c) 3
(d) 4

Question. From a point P( λ,λ,λ), perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = – x, z = – 1. If P is such that ∠QPR is a right angle, then the possible value(s) of λ is: (are)         C
(a) 2
(b) 1
(c) –1
(d) − 2

Question. The (d)r.’s of normal to the plane through (1, 0, 0) and (0,1, 0) which makes an angle π / 4 with plane x + y = 3, are:
(a) 1, √2,1
(b) 1,1, √2
(c) 1, 1, 2
(d) √2,1,1

Question. The angle between the planes 3x − 4y + 5z = 0 and 2x − y − 2z = 5 is:
(a) π/3
(b) π/2
(c) π/6
(d) None of these

Question. The point of intersection of the lines x–5/3 = y-7/-1 = z+2/1 , x+3/-36 = y-3/2 = z-6/4 is:
(a) 21 , 5/3 , 10/3
(b) ( 2,10, 4)
(c) (−3, 3, 6)
(d) (5, 7, − 2)

Question. If a plane cuts off intercepts –6, 3, 4 from the co-ordinate axes, then the length of the perpendicular from the origin to the plane is:
(a) 1/√61
(b) 13/√61
(c) 12/√29
(d) 5/√41

Question. The equation of the plane containing the line of intersection of the planes 2x − y = 0 and y −3z = 0 and perpendicular to the plane 4x + 5y −3z −8 = 0 is:
(a) 28x −17y + 9z = 0
(b) 28x +17y + 9z = 0
(c) 28x −17y + 9x = 0
(d) 7x −3y + z = 0

Question. A point moves so that its distances from the points (3, 4, –2) and (2, 3, – 3) remains equal. The locus of the point is:
(a) A line
(b) A plane whose normal is equally inclined to axes
(c) A plane which passes through the origin
(d) A sphere

Question. 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, –1, 5) is given by:
(a) x + 5y − 6z +19 = 0
(b) x −5y + 6z −19 = 0
(c) x + 5y + 6z +19 = 0
(d) x −5y − 6z −19 = 0

Question. Value of k such that the line x–1/2 = y–1/3 = z-k/k , perpendicular to normal to the plane r(2i + 3j+ 4k) = 0 is:
(a) –13/4
(b) –17/4
(c) 4
(d) 5

Question. The value of k for which the planes 3x − 6y − 2z = 7 and 2x + y − kz = 5 are perpendicular to each other, is:
(a) 0
(b) 1
(c) 2
(d) 3

Question. The equation of line of intersection of the planes 4x + 4y −5z =12 , 8x +12y −13z = 32 can be written as: 

Question. The equation of the plane containing the two lines 

(a) 8x + y −5z − 7 = 0
(b) 8x + y + 5z − 7 = 0
(c) 8x − y −5z − 7 = 0
(d.) None of these

Question. The point at which the line joining the points (2, –3, 1) and (3, –4, –5) intersects the plane 2x + y + z = 7 is:
(a) (1, 2, 7)
(b) (1, –2, 7)
(c) (–1, 2, 7)
(d) (1, –2, –7)

Question. If OABC is a tetrahedron such that OA²+ BC²= OB² + CA²= OC² + AB² then:     
(a) OA ⊥ BC
(b) OB ⊥ CA
(c) OC ⊥ AB
(d) AB ⊥ BC

Question. Which of the following is/are true?
I. The reflection of the point (α, β, γ) in the xy-plane is (α, β, –γ) .
II. The locus represented by xy + yz = 0 is a pair of perpendicular lines.
III. The line r = 2î – 3ĵ- kˆ + λ (î – ĵ + 2kˆ ) lies in the plane r = (3î + ĵ – kˆ ) + 2 = 0 . 
(a) Only I is true
(b) I and II are true
(c) I and III are true
(d) II and III are true

Question. The plane passing through the point (–2, –2, 2) and containing the line joining the points (1, 1, 1) and (1, –1, 2) makes intercepts on the coordiantes axes, the sum of whose lengths is
(a) 3
(b) – 4
(c) 6
(d) 12

Question. If the points (1, 1, p) and (–3, 0, 1) be equidistant from the plane r.(3î + 4ĵ – 12k̂) + 13 = 0 , then the value of p is
(a) 3/7
(b) 7/3
(c) 4/3
(d) 3/4

Question. If two points are P (7, –5, 11) and Q (–2, 8, 13), then the projection of PQ on a straight line with direction cosines 1/3 , 2/3 , 2/3 is
(a) 1/2
(b) 26/3
(c) 4/3
(d) 7

Assertion – Reason Type Questions :

(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

Question. Assertion : Distance of a point with position vector a from a plane r. N = d is given by |a . N – d|.
Reason : The length of perpendicular from origin O to the

Question. Assertion: The pair of lines given by r̅ = î – ĵ + λ (2î + k) and r̅ = 2î – k̂+ μ (î + ĵ – k̂) intersect.
Reason: Two lines intersect each other, if they are not parallel and shortest distance = 0.

Question. Consider the planes 3x – 6y – 2z = 15 and 2x + y – 2z = 5.
Assertion : The parametric equations of the line of intersection of the given planes are x = 3 + 14t, y = 1 + 2t, z = 15t.
Reason : The vector 14î + 2ĵ +15kˆ is parallel to the line of intersection of given planes.

Question. Consider three planes
P₁ : x – y + z = 1
P₂ : x + y – z = 1
P₃ : x – 3y + 3z = 2
Let L₁, L₂, L₃ be the lines of intersection of the planes P₂ and P3, P₃ and P₁, P₁ and P₂, respectively.
Assertion : At least two of the lines L₁, L₂ and L₃ are nonparallel
Reason : The three planes doe not have a common point.

Question. Consider the lines
L1 : x+1/3 = y+2/1 = z+1/2 , L2 : x-2/2 = y-2/2 = z-3/3 , 
Assertion: The distance of point (1, 1, 1) from the plane passing through the point (–1, –2, –1) and whose normal is perpendicular to both the lines L₁ and L₂ is 13/5√3 .
Reason: The unit vector perpendicular to both the lines L₁

Question. Assertion: If a variable line in two adjacent positions has direction cosines l, m, n, and l + δl, m + δm, n + δn, then the small angle δθ between the two positions is given by δθ = δl² + δm² + δn²
Reason: If O is the origin and A is (a, b, c), then the equation of plane through at right angle to OA is given by ax + by + cz = a² + b² + c².

Question. The equation of the line passing through (–4, 3, 1), parallel to the plane x + 2y – z – 5 = 0 and intersecting the line (x + 1)/-3 = (y – 3)/2 = (z – 2)/-1 is :
(image 80)

Question. The distance of the point (1, – 5, 9) from the plane x – y + z = 5 measured along a straight x = y = z is
(a) 10√3
(b) 5√3
(c) 3√10
(d) 3√5

Question. If x = a, y = b, z = c is a solution of the system of linear equations
x + 8y + 7z = 0
9x + 2y + 3z = 0
x + y + z = 0
such that the point (a, b, c) lies on the plane x + 2y + z = 6, then 2a + b + c equals :
(a) –1
(b) 0
(c) 1
(d) 2

Question. Statement -1 : The point A(3, 1, 6) is the mirror image of the point B(1, 3, 4) in the plane x – y + z = 5.
Statement -2: The plane x – y + z = 5 bisects the line segment oining A(3, 1, 6) and B(1, 3, 4).
(a) Statement -1 is true, Statement -2 is true ; Statement – 2 is not a correct explanation for Statement -1.
(b) Statement -1 is true, Statement -2 is false.
(c) Statement -1 is false, Statement -2 is true .
(d) Statement – 1 is true, Statement 2 is true ; Statement -2 is a correct explanation for Statement -1.

Question. Let a plane P contain two lines r̅ = î + λ(î + ĵ), λ ∈ R and r̅ = – ĵ + μ( ĵ – k ), m ∈ R. If Q(α, β, γ) is the foot of the perpendicular drawn from the point M(1, 0, 1) to P, then 3(α + β + γ) equals ____________.

Question. 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
(a) any value
(b) exactly one value
(c) exactly two values
(d) exactly three values

Question. The shortest distance between the lines (x – 3)/3 = (y – 8)/-1 = (z – 3)/1 and (x+3)/-3 = (y + 7)/2 = (z – 6)/4 is :
(a) 2√30
(b) (7/2)√30
(c) 3√30
(d) 3

Question. The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also through the point :
(a) (0, 6, –2)
(b) (–2, 0, 1)
(c) (0, –6, 2)
(d) (2, 0, –1)

Question. The plane containing the line (x – 1)/1 = (y – 2)/2 = (z -3)/3 and parallel to the line x/1 = y/1 = z/4 passes through the point:
(a) (1, – 2, 5)
(b) (1, 0, 5)
(c) (0, 3, –5)
(d) (– 1, – 3, 0)

Question. The line L given by x/5 +y/b = 1 passes through the point (13, 32). The line K is parallel to L and has the equation x/c + y/3 = 1. Then the distance between L and K is
(a) √17
(b) 17/√15
(c) 23/√17
(d) 23/√15

Question. If L1 is the line of intersection of the planes 2x – 2y + 3z – 2 = 0, x – y + z +1= 0 and L2 is the line of intersect ion of the planes x + 2y – z – 3 = 0, 3x – y + 2z -1= 0 , then the distance of the origin from the plane, containing the lines L₁ and L₂, is :
(a) 1/3√2
(b) 1/2√2
(c) 1/√2
(d) 1/4√2

Question. The perpendicular distance from the origin to the plane containing the two lines, (x+2)/3 = (y – 2)/5 = (z+5) and (x – 1)/1 = (y – 4)/4 = (z+4)/7, is :
(a) 11√6
(b) 11/√6
(c) 11
(d) 6√11

Question. Let the line (x – 2)/3 = (y – 1)/-5 = (z+2)/2 lie in the plane x + 3y – αz + β = 0. Then (α, β) equals
(a) (–6, 7)
(b) (5, –15)
(c) (–5, 5)
(d) (6, –17)

Question. The distance of the point (1, 0, 2) from the point of intersection of the line (x – 2)/3 = (y+1)/4 = (z – 2)/12 and the plane x – y + = 16, is
(a) 3√21
(b) 13
(c) 2√14
(d) 8

Question. The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an angle π/4 with the plane y – z + 5 = 0 are :
(a) 2, –1, 1
(b) 2, √2,- 2
(c) √2, 1, -1
(d) 2√3,1, -1

Question. If the line (x – 2)/3 = (y + 1)/2 = (z – 1)/-1 intersects the plane 2x + 3y – z + 13 = 0 at a point P and the plane 3x + y + 4z = 16 at a point Q, then PQ is equal to:
(a) 14
(b) √14
(c) 2√7
(d) 2√14

Question. If the points (1, 1, λ) and (–3, 0, 1) are equidistant from the plane, 3x + 4y – 12z + 13 = 0, then λ satisfies the equation :
(a) 3x² + 10x – 13 = 0
(b) 3x² – 10x + 21 = 0
(c) 3x² – 10x + 7 = 0
(d) 3x² + 10x – 7 = 0

Question. The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to y-axis also passes through the point:
(a) (– 3, 0, – 1)
(b) (– 3, 1, 1)
(c) (3, 3, – 1)
(d) (3, 2, 1)

Question. The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line x/2 = y/3 = z/-6 is :
(a) 7/5
(b) 1
(c) 1/7
(d) 7

Question. The coordinates of the foot of the perpendicular from the point (1, –2, 1) on the plane containing the lines, (x+1)/6 = (y – 1)/7 = (z – 3)/8 and (x – 1)/3 = (y – 2)/5 = (z – 3)/7, is :
(a) (2, –4, 2)
(b) (–1, 2, –1)
(c) (0, 0, 0)
(d) (1, 1, 1)

Question. The line of intersection of the planes r̅.(3î – + k̂ ) = 1 and r̅.( î + 4 – 2k̂ ) = 2, îs :
(image 94)

Question. The number of distinct real values of λ for which the lines (x – 1)/1 = (y – 2)/2 = (z+3)/λ² and (x – 3)/1 = (y – 2)/λ² = (z – 1)/2 are coplanar is :
(a) 2
(b) 4
(c) 3
(d) 1

Question. If the shortest distance between the lines (x – 1)/α = (y + 1)/-1 = z/1, (α ≠ -1) and x + y + z +1 = 0 = 2x – y + z + 3 is 1/√3 then a value α is :
(a) – 16/19
(b) – 19/16
(c) 32/19
(d) 19/32

Question. 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 co-ordinates of each of the points of intersection are given by
(a) (2a,3a,3a), (2a,a,a)
(b) (3a, 2a,3a), (a, a, a)
(c) (3a,2a,3a), (a, a, 2a)
(d) (3a,3a,3a), (a,a,a)

Question. If the distance between planes, 4x – 2y – 4z + 1 = 0 and 4x – 2y – 4z + d = 0 is 7, then d is:
(a) 41 or – 42
(b) 42 or – 43
(c) – 41 or 43
(d) – 42 or 44

Question. Two lines (x – 3)/1 = (y+1)/3 = (z – 6)/-1 and (x + 5)/7 = (y – 2)/-6 = (z – 3)/4 intersect at the point R. The reflection of R in the xy- plane has coordinates :
(a) (2, –4, –7)
(b) (2, 4, 7)
(c) (2, –4, 7)
(d) (–2, 4, 7)

Question. The sum of the intercepts on the coordinate axes of the plane passing through the point (– 2, – 2, 2) and containing the line oining the points (1, – 1, 2) and (1, 1, 1) is
(a) 12
(b) – 8
(c) – 4
(d) 4

Question. The distance of the point (1, –2, 4) from the plane passing through the point (1, 2, 2) and perpendicular to the planes x – y + 2z = 3 and 2x – 2y + z + 12 = 0, is :
(a) 2
(b) √2
(c) 2√2
(d) 1/√2

Question. A symmetrical form of the line of intersection of the planes x = ay + b and = cy + d is
(image 107)

Question. The equation of a plane through the line of intersection of the planes x + 2y = 3, y –2 + 1= 0, and perpendicular to the first plane is :
(a) 2x – y – 10z = 9
(b) 2x – y + 7z = 11
(c) 2x – y + 10z = 11
(d) 2x – y – 9z = 10

Question. A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0, passes through the point :
(a) (1, –4, 1)
(b) (1, 4, –1)
(c) (2, 4, 1)
(d) (2, –4, 1)

Question. If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of ΔABC is :
(a) 1/x² + 1/y² + 1/z² = 1
(b) 1/x² + 1/y² + 1/z² = 3
(c) 1/x² + 1/y² + 1/z² = 1/9
(d) 1/x² + 1/y² + 1/z² = 9

Question. Let Q be the foot of perpendicular from the origin to the plane 4x – 3y + z + 13 = 0 and R be a point (– 1, – 6) on the plane. Then length QR is :
(a) √14
(b) √(19/2)
(c) 3√(7/2)
(d) 3/√2

Question. A vector n̅ is inclined to x-axis at 45 , to y-axis at 60 and at an acute angle to z-axis. If n̅is a normal to a plane passing through the point (√2,-1,1) then the equation of the plane is :
a) 4√2x + 7y + z – 2
(b) 2x + y + 2z = 2√2 +1
(c) 3√2x – 4y -3z = 7
(d) √2x – y – z = 2

Question. The distance of the point -î + 2ĵ + 6k̂ from the straight line that passes through the point 2î + 3ĵ – 4k̂ and is parallel to the vector 6î + 3ĵ – 4k̂ is
(a) 9
(b) 8
(c) 7
(d) 10

Question. A plane bisects the line segment oining the points (1, 2, 3) and (– 3, 4, 5) at right angles. Then this plane also passes through the point.
(a) (– 3, 2, 1)
(b) (3, 2, 1)
(c) (1, 2, – 3)
(d) (– 1, 2, 3)

Question. Consider the following planes
P : x + y – 2z + 7 = 0
Q : x + y + 2z + 2 = 0
R : 3x + 3y – 6z – 11 = 0
(a) P and R are perpendicular
(b) Q and R are perpendicular
(c) P and Q are parallel
(d) P and R are parallel

Question. If the angle between the lines, x/2 = y/2 = z/1 and (5 – x)/-2 = (7y – 14)/P = (z – 3)/4 is cos^-1(2/3), then P is equal to
(a) – 7/4
(b) 2/7
(c) – 4/7
(d) 7/2

Question. The values of a for which the two points (1, a, 1) and (– 3, 0, a) lie on the opposite sides of the plane 3x + 4y – 12z + 13 = 0, satisfy
(a) 0 < a < 1/3
(b) – 1 < a < 0
(c) a < – 1 or a < 1/3
(d) a = 0

Question. The mirror image of the point (1, 2, 3) in a plane is (- 7/3, – 4/3, – 1/3). Which of the following points lies on this plane?
(a) (1, 1, 1)
(b) (1, –1, 1)
(c) (–1, –1, 1)
(d) (–1, –1, –1)