Buds Public School , Dubai
Mathematics – Three Dimensional Geometry
Grade 12 S
1. The points A(4,5,10) , B(2,3,4) and C(1,2,-1) are the vertices of a parallelogram ABCD . Find the vector
and Cartesian equations for the sides AB and BC and find the coordinates of D .
𝑥−4
Ans :
1
=
𝑦−5
1
=
𝑧−10
3
𝑥−2
,
1
=
𝑦−3
1
=
𝑧−4
5
, (3,4,5)
2. Find the vector equation of a line passing through a point with position vector 2 𝑖⃗ − 𝑗 + ⃗⃗⃗𝑘 and parallel
to the line joining the points , −⃗⃗𝑖 + 4𝑗 +⃗⃗⃗𝑘 , and ⃗⃗𝑖 + 2𝑗 + 2⃗⃗⃗𝑘. Also find the Cartesian equivalent of this
equation .
𝑥−2
Ans :
2
𝑦+1
=
−2
𝑧−1
=
1
3. The Cartesian equation of a line 6𝑥 − 2 = 3𝑦 + 1 = 2𝑧 − 2 . Find its direction ratios and also find
1
1
⃗ )
vector equation of the line .
Ans : ⃗⃗𝑟 = 𝑖⃗ − 𝑗 + ⃗⃗⃗𝑘 + 𝜇( ⃗⃗𝑖 + 2𝑗 + 3𝑘
3
4. Find the point on the line
𝑥+2
𝑦+1
=
3
𝑧−3
=
2
2
3
at a distance of 3√2 from the point (1,2,3) .
56
43
Ans : (-2,-1,3), (17 , 17 ,
5. Find the point on the line
𝑥+2
3
=
𝑦+1
2
𝑧−3
=
2
111
17
)
at a distance of 5 units from the point (1,3,3).
Ans : (4,3,7) , (-8,-5,-1)
6. Show that the points whose position vectors are −2 𝑖 + 3𝑗 , 𝑖⃗ + 2𝑗 + 3⃗⃗⃗𝑘 , and 7 𝑖⃗ − ⃗⃗⃗𝑘 are collinear .
7. Find the Cartesian and vector equation of a line which passes through the point (1,2,3) and is parallel to
−𝑥−2
the line
=
3
𝑦+3
7
2𝑧−6
=
3
.
Ans :
𝑥−1
−2
𝑦−2
=
14
=
𝑧−3
3
, 𝑟⃗⃗ = 𝑖⃗ + 2𝑗 + 3⃗⃗⃗𝑘 + 𝜇(−2𝑖⃗ + 14𝑗 + 3⃗⃗⃗𝑘 )
8. The Cartesian equation of a line 3𝑥 + 1 = 6𝑦 − 2 = 1 − 𝑧. Find the fixed point through which it
passes , its direction ratios and also find vector equation of the line .
−1 1
1
1
⃗ )
Ans : ( , , 1) , (2,1, −6), 𝑟⃗⃗ =− 𝑖⃗ + 𝑗 + ⃗⃗⃗⃗⃗⃗⃗⃗
𝑘 + 𝜇( ⃗⃗⃗⃗⃗
2𝑖 + 𝑗 − 6𝑘
3
3
3
3
𝑥
9. Find the equation of the line passing through the point (−1,3, −2) and perpendicular to the lines 1 =
𝑦
2
=
𝑧
2
,
𝑥+2
−3
𝑦−1
=
2
𝑧+1
=
5
.
Ans :
−𝑥+2
10. Find the angle between the pair of lines
−2
=
𝑦−1
7
=
𝑧+3
−3
,
𝑥+1
2
𝑥+2
−1
=
=
𝑦−3
=
−7
2𝑦−8
4
=
𝑧+2
4
𝑧−5
4
𝜋
. Ans : 2 .
11. Find the value of 𝜇 so that the following lines are perpendicular to each other .
𝑥−5
5𝜇+2
=
2−𝑦
5
=
1−𝑧
𝑥
,
−1
1
𝑥−2
12. Show that the line
=
=
2
2𝑦+1
4𝜇
𝑦−1
3
=
=
1−𝑧
−3
𝑧−3
4
.
𝑎𝑛𝑑
Ans : 𝜇 = 1
𝑥−4
5
=
𝑦−1
2
= 𝑧 intersect . Find their point of intersection .
Ans : (-1,-1,-1)
13. Determine whether the following pair of lines intersect or not :
𝑥−5
4
=
𝑦−7
4
=
𝑧+3
−5
,
𝑥−8
7
=
𝑦−4
1
=
𝑧−5
3
Ans : Yes
𝑥
14. Find the image of the point (1,6,3) in the line 1 =
𝑦−1
2
=
𝑧−2
3
. Also write the equation of the line
joining the given point and its image and find the length of the segment joining the given point and its
image .
𝑥−1
Ans : (1,0,7) ,
𝑦−6
=
0
−6
=
𝑧−3
4
, 2√13
15. Find the perpendicular distance of the point (1,0,0) from the line
𝑥−1
2
𝑦+1
=
−3
=
𝑧+10
8
. Also find the
coordinates of the foot of the perpendicular and the equation of the perpendicular .
Ans : 2√6 ,(3, −4, −2), 𝑟⃗⃗ = 𝑖⃗ + 𝜇( 𝑖⃗ − 2𝑗 − ⃗⃗⃗𝑘 ).
16. Find the Shortest distance between the following lines :
a) ⃗⃗𝑟 = 𝑖⃗ + 2𝑗 + 3⃗⃗⃗𝑘 + 𝜇(2𝑖⃗ + 3𝑗 + 4⃗⃗⃗𝑘 ) 𝑎𝑛𝑑 , ⃗⃗𝑟 = ⃗⃗⃗⃗
2𝑖 + 4𝑗 + 5⃗⃗⃗𝑘 + 2𝛼(4𝑖⃗ + 6𝑗 + 8⃗⃗⃗𝑘 )
b)
𝑥−1
=
1
𝑦−5
−2
=
𝑧−7
1
𝑎𝑛𝑑
𝑥+1
7
=
𝑦+1
−6
𝑧+1
=
√5
Ans : a)
1
√29
𝑏) 2 √29
17. Write the vector equation of the following lines and hence determine the distance between them .
𝑥−1
2
=
𝑦−2
3
=
𝑧+4
6
,
𝑥−3
4
=
𝑦−3
6
=
𝑧+5
12
√293
Ans : 𝑟⃗⃗ = 𝑖⃗ + 2𝑗 − 4⃗⃗⃗𝑘 + 𝜇(2𝑖⃗ + 3𝑗 + 6⃗⃗⃗𝑘 ) , ⃗⃗𝑟 = ⃗⃗⃗⃗
3𝑖 + 3𝑗 − 5⃗⃗⃗𝑘 + 𝛼(4𝑖⃗ + 6𝑗 + 12⃗⃗⃗𝑘 ), 7
18. The Cartesian equation of a line AB is
2𝑥−1
√3
=
𝑦=2
2
=
3
. Find the direction cosines of a line parallel
Ans : (
to AB .
19. If the equation of the line AB are
Ans :
𝑧−3
3−𝑥
1
=
𝑦+2
2
=
𝑧−3
3
√5
,
√55
4
√55
,
6
)
√55
, write the direction ratios of a line parallel to AB .
(−1,2, −4)
20. Write the vector equation of a line given by
𝑥−5
3
=
𝑦+4
7
=
𝑧−6
2
.
Ans : 𝑟⃗⃗ = ⃗⃗⃗⃗
5𝑖 − 2𝑗 + 3⃗⃗⃗𝑘 + 𝜇(𝑖⃗ + 2𝑗 + 3⃗⃗⃗𝑘 )
Planes :
1. Find the equation of the plane passing through the point (1,2,1) and perpendicular to the line joining the
points (1,4,2) and (2,3,5) . Find also the perpendicular distance of the origin from this plane .
Ans : 𝑥 − 𝑦 + 3𝑧 − 2 = 0,
2
√11
2. Find the equation of the plane passing through the point (1,1,-1) and perpendicular to the planes 𝑥 +
2𝑦 + 3𝑧 − 7 = 0 𝑎𝑛𝑑 2𝑥 − 3𝑦 + 4𝑧 = 0 .
Ans : 17𝑥 + 2𝑦 − 7𝑧 = 26
3. Find the equation of the plane passing through the point (-1,-1,2) and perpendicular to the planes 𝑥 +
2𝑦 + 2𝑧 = 5 𝑎𝑛𝑑 3𝑥 + 3𝑦 + 2𝑧 = 8 .
Ans : 2𝑥 − 4𝑦 + 3𝑧 − 8 = 0
4. Obtain the equation of the plane passing through the point (1,-3,2) and perpendicular to the planes 3𝑥 +
2𝑦 − 3𝑧 = 1 𝑎𝑛𝑑 5𝑥 − 4𝑦 + 𝑧 = 5 .
Ans : 5𝑥 + 9𝑦 + 11𝑧 − 8 = 0
4. Find the equation of the plane passing through the points (-1,-1,2) and (2,-2,2) and which
perpendicular to the plane 6𝑥 − 2𝑦 + 2𝑧 = 9 .
Ans : 𝑥 + 𝑦 − 2𝑧 + 4 = 0
5. Find the equation of a plane which is perpendicular to the plane 5𝑥 + 3𝑦 + 6𝑧 + 8 = 0 and which
contains the line of intersection of the planes 𝑥 + 2𝑦 + 3𝑧 − 4 = 0 𝑎𝑛𝑑 2𝑥 + 𝑦 − 𝑧 + 5 = 0 .
Ans : 51𝑥 + 15𝑦 − 50𝑧 + 173 = 0
6. Find the Cartesian and Vector equation of the planes through the intersection of the planes
a) ⃗⃗𝑟 . (2𝑖⃗ + 6𝑗) + 12 = 0 and 𝑟⃗⃗ . ( 3𝑖⃗ − 𝑗 + 4⃗⃗⃗𝑘 ) = 0 which are at a unit distance from the origin .
⃗ ) + 3 = 0 and 𝑟⃗⃗ . (−1𝑖⃗ + 2𝑗 − 2⃗⃗⃗𝑘 ) + 3 = 0 ) : λ= ±2
Ans : 𝑟⃗⃗ . (2𝑖⃗ + 𝑗 + 2𝑘
b) ⃗⃗𝑟 . (𝑖⃗ + 3𝑗) + 6 = 0 and 𝑟⃗⃗ . ( 3𝑖⃗ − 𝑗 − 4⃗⃗⃗𝑘 ) = 0 which are at a unit distance from the origin .
⃗ ) + 6 = 0 or 𝑟⃗⃗ . (4𝑖⃗ + 2𝑗 − 4⃗⃗⃗𝑘) + 6 = 0 )
Ans : 𝑟⃗⃗ . (−2𝑖⃗ + 4𝑗 + 4𝑘
7. Find the equation of the plane that contains the line of intersection of the planes
⃗ ) − 4 = 0 and 𝑟⃗⃗ . ( 2𝑖⃗ + 𝑗 − 4⃗⃗⃗𝑘 )+5 = 0 which is perpendicular to the plane
𝑟⃗⃗ . (𝑖⃗ + 2𝑗 + 2𝑘
𝑟⃗⃗ . ( 5𝑖⃗ + 3𝑗 − 6⃗⃗⃗𝑘 ) + 8 = 0.
Ans : 33𝑥 + 45𝑦 − 50𝑧 − 41 = 0
8. Find the equation of the plane passing through the intersection of the planes ⃗⃗𝑟 . ( 2𝑖⃗ + 𝑗 + 3⃗⃗⃗𝑘) = 7
and 𝑟⃗⃗ . ( 2𝑖⃗ + 5𝑗 + 3⃗⃗⃗𝑘) = 9 and the point (2,1,3)
Ans : 𝑟⃗⃗ . ( 2𝑖⃗ − 13𝑗 + 3⃗⃗⃗𝑘) = 0
9. Find the distance between the point (6,5,9) and the plane determined by the points A(3,-1,2) , B(5,2,4)
and C(-1,-1,6)
Ans : 3𝑥 − 4𝑦 + 3𝑧 = 19 ,
6
√34
⃗ ) + λ( 𝑖⃗ − 𝑗 + 4𝑘
⃗ ) is parallel to the
10 . Show that the line whose vector equation is 𝑟⃗⃗ = (2 𝑖⃗ − 2𝑗 + 3𝑘
plane whose vector equation is 𝑟⃗⃗ . ( 𝑖⃗ + 5𝑗 + ⃗⃗⃗𝑘) = 5. Also find the distance between them .
𝐴𝑛𝑠 ∶
6
√34
11. Find the equation of the plane passing through the line of intersection of the planes
2𝑥 + 𝑦 − 𝑧 = 3 , 5𝑥 − 3𝑦 + 4𝑧 + 9 = 0 and parallel to the line
𝑥−1
2
=
𝑦−3
4
=
𝑧−5
5
.
Ans : 7𝑥 + 9𝑦 − 10𝑧 − 27 = 0
12. Find the equation of the plane passing through the line of intersection planes 𝑟⃗⃗ . ( 𝑖⃗ + 𝑗 + ⃗⃗⃗𝑘) = 1 and
𝑟⃗⃗ . ( 2𝑖⃗ + 3𝑗 − ⃗⃗⃗𝑘) + 4 = 0 and parallel to x-axis . Ans : ⃗⃗𝑟 . (−𝑗 + 3⃗⃗⃗𝑘) = 6.
_________________________________
the 𝑎
⃗⃗⃗ =2 𝑖⃗ + 2𝑗 + 3⃗⃗⃗𝑘 , ⃗⃗⃗
𝑏 = −⃗⃗⃗𝑖 + 2𝑗 +⃗⃗⃗𝑘 and 𝑐⃗⃗⃗ = 3 𝑖⃗ + 𝑗