3- Dimensional geometry:
1) Show that the points A (2,3,-4), B(1, -2,3) and C(3, 8,-11) are collinear.
𝑥−2
2) Find the angle between the lines:
=
2
3) Find the values of 𝑝 so that the lines
1−𝑥
3
𝑦−1
5
=
=
𝑧+3
and
𝑥+2
=
𝑦−4
−3
−1
8
7𝑦−14
𝑧−3
7−7𝑥
2𝑝
=
2
and
3𝑝
=
=
𝑧−5
4
𝑦−5
1
=
6−𝑧
5
are at
right angle.
4) Find the shortest distance between the lines 𝑟⃗ = (𝑖̂ + 2𝑗̂ + 𝑘̂ ) + µ(𝑖̂ − 𝑗̂ + 𝑘̂ ) and
̂ ).
𝑟⃗ = (2𝑖̂ − 𝑗̂ + 𝑘̂ ) + 𝜕(2𝑖̂ + 𝑗̂ + 2𝑘
5) Find the shortest distance between the lines whose vector equations are 𝑟.
⃗⃗⃗ =
(1 − 𝑡 )𝑖̂ + (𝑡 − 2)𝑗̂ + (3 − 2𝑡 )𝑘̂ and 𝑟.
⃗⃗⃗ = (𝑠 + 1)𝑖̂ + (2𝑠 − 1)𝑗̂ − (2𝑠 + 1)𝑘̂
6) Find the coordinates of the foot of the perpendicular drawn from the origin to the
plane 2𝑥 − 3𝑦 + 4𝑧 − 6 = 0
7) Find the vector equations of the plane passing through the intersection of the
̂ + 4𝑘̂ ) = −5 and the points (1, 1, 1)
planes 𝑟⃗. (𝑖̂ + 𝑗̂ + 𝑘̂ ) = 6 and 𝑟⃗. (2𝑖̂ + 3𝑗
8) Find the angle between the line
𝑥+1
2
=
𝑦
3
=
𝑧−3
6
and the plane 10𝑥 + 2𝑦 − 11𝑧 = 3
9) Find the equation of the plane through the line of intersection of the planes 𝑥 + 𝑦 +
𝑧 = 1 and 2𝑥 + 3𝑦 + 4𝑧 = 5 which is perpendicular to the plane 𝑥 − 𝑦 + 𝑧 = 0
10) A line makes angles 𝛼, 𝛽, 𝛾 and 𝛿 with the diagonals of a cube, prove that 𝑐𝑜𝑠 2 𝛼 +
𝑐𝑜𝑠 2 𝛽 + 𝑐𝑜𝑠 2 𝛾 + 𝑐𝑜𝑠 2 𝛿 =
4
3
11) Find the equation of the plane that contains the point (1, −1, 2) and is
perpendicular to each of the planes 2𝑥 + 3𝑦 − 2𝑧 = 5 and 𝑥 + 2𝑦 − 3𝑧 = 8
12) Find the distance between the point P(6,5,9) and the plane determined by the
points A(3, -1, 2), B(5, 2, 4) and C(-1, -1, 6)
13) Find the coordinates of the point where the line through the points A(3, 4, 1) and
B(5, 1, 6) crosses the XY-plane.
14) Find the coordinates of the point where the line through (3, -4, -5) and ( 2, -3, 1)
crosses the planes 2𝑥 + 𝑦 + 𝑧 = 7
15) Find the equation of the plane passing through the point(-1, 3, 2) and perpendicular
to each of the planes 𝑥 + 2𝑦 + 3𝑧 = 5 and 3𝑥 + 3𝑦 + 𝑧 = 0
16) Find the points (1,1, 𝑝) and ( -3, 0, 1) be equidistant from the plane 𝑟⃗. (3𝑖̂ + 4𝑗̂ −
12𝑘̂ ) + 13 = 0, then find the value of 𝑝
Prepared by Mr. Anirban Nayak
PGT, Mathematics, DPS Jodhpur
Mobile No.- 9828353006
17) If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the
plane passing through P and perpendicular to OP.
18) Find the equation of the plane which contains the line of intersection of the planes
𝑟⃗. (𝑖̂ + 2𝑗̂ + 3𝑘̂ ) − 4 = 0, 𝑟⃗. (2𝑖̂ + 𝑗̂ − 𝑘̂ ) = −5 and which is perpendicular to the
plane 𝑟⃗. (5𝑖̂ + 3𝑗̂ − 6𝑘̂ ) = −8
19) Find the distance of the point (-1, -5, -10) from the point of intersection of the line
̂ ) + µ(3𝑖̂ + 4𝑗̂ + 2𝑘̂ ) and the plane 𝑟⃗. (𝑖̂ − 𝑗̂ + 𝑘̂ ) = 5.
𝑟⃗ = (2𝑖̂ − 𝑗̂ + 2𝑘
20) Find the vector equation of the line passing through (1, 2, 3) and parallel to the
planes 𝑟⃗. (𝑖̂ − 𝑗̂ + 2𝑘̂ ) = 5 and 𝑟⃗. (3𝑖̂ + 𝑗̂ + 𝑘̂ ) = 6
21) Prove that if a plane has the intercepts 𝑎, 𝑏, 𝑐 and is at a distance of 𝑝 units from the
origin, then
1
𝑎2
+
1
𝑏2
+
1
𝑐2
=
1
𝑝2
.
EXTRA PRoblems
1) Find the equation of the plane passing through the point (-1, 2, 1) and perpendicular
to the line joining the points (-3, 1, 2) and ( 2, 3, 4). Also , find the perpendicular
distance of the origin from this plane.
2) Find the distance between the point P(6,5,9) and the plane determined by the
points A(3, -1, 2), B(5, 2, 4) and C(-1, -1, 6)
3) Find the perpendicular distance of the point (1, 0, 0) from the line
4) Find the point on the line
𝑥+2
3
=
𝑦+1
2
=
𝑧−3
2
𝑥−1
2
=
𝑦+1
−3
=
𝑧+10
8
at a distance 3√2 from the point ( 1, 2, 3)
5) Find the coordinates of the foot of perpendicular and length of the perpendicular
drawn from the point P(5, 4,2) to the line 𝑟⃗ = −𝑖̂ + 3𝑗̂ + 𝑘̂ + µ(2𝑖̂ + 3𝑗̂ − 𝑘̂ ). Also
find the image of P in the line.
𝑥
𝑦−1
1
𝑧−3
2
6) Find the image of the point (1, 6, 3) in the line =
7) If the lines
𝑥−1
−3
=
𝑦−2
−2𝑘
=
𝑧−3
2
and
𝑥−1
𝑘
=
𝑦−2
1
=
5
=
𝑧−2
3
.
are perpendicular, find the value of
𝑘 and hence find the equation of plane containing these lines.
8) Find the image of the point having position vector 𝑖̂ + 3𝑗̂ + 4𝑘̂ in the plane
𝑟⃗. (2𝑖̂ − 𝑗̂ + 𝑘̂ )+3 =0
Prepared by Mr. Anirban Nayak
PGT, Mathematics, DPS Jodhpur
Mobile No.- 9828353006
9) Find the distance of the point ( -2, 3, -4) from the line
𝑥+2
3
=
2𝑦+3
4
=
3𝑧+4
5
measured
parallel to the plane 4𝑥 + 12𝑦 − 3𝑧 + 1 = 0
10) Show that the lines
𝑥+3
−3
=
𝑦−1
1
𝑧−5
=
5
,
𝑥+1
=
−1
𝑦−2
2
=
𝑧−5
5
are coplanar. Also find the
equation of the plane containing the lines.
11) Find the distance of the point( 1, -2, 3) from the plane 𝑥 − 𝑦 + 𝑧 = 5, measured
parallel to the line
𝑥−1
2
=
𝑦−3
3
=
𝑧+2
−6
12) Find the equation of two lines through the origin which intersect the lines
𝑦−3
1
𝑧
𝜋
1
3
𝑥−3
2
=
= at angles of each.
13) Prove that the points P(1, -1, 3) and Q(3,3,3) are equidistant from the plane
𝑟⃗. (5𝑖̂ + 2𝑗̂ − 7𝑘̂ )+9=0. Also find the position vector of the point at which the line
joining P, Q intersects the plane.
14) Find the distance of the point (2, 3, 4) from the plane 3𝑥 + 2𝑦 + 2𝑧 + 5 = 0,
measured parallel to the line
𝑥+3
3
=
𝑦−2
6
=
𝑧
2
15) Find the image of the point P(3, 5, 7) in the plane 2𝑥 + 𝑦 + 𝑧 = 16.
16) Find the Cartesian as well as the vector equation of the planes passing through the
intersection of the planes 𝑟⃗. (2𝑖̂ + 6𝑗̂) + 12 = 0 and 𝑟⃗. (3𝑖̂ − 𝑗̂ + 4𝑘̂ ) = 0, which
are at unit distance from the origin.
17) Show that the lines
𝑥−1
3
=
𝑦+1
2
=
𝑧−1
5
and
𝑥+2
4
=
𝑦−1
3
=
𝑧+1
−2
do not intersect.
18) A plane meets the coordinate axes in A, B, C such that the centroid of triangle ABC is
𝑥
𝑦
𝑧
𝑎
𝑏
𝑐
the point (p,q, r).Show that the equation of the plane is + + = 3
19) Find the equation of the 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
20) A variable plane which remains at a constant distance 3𝑝 from the origin cut the
coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is
𝑥 −2 + 𝑦 −2 + 𝑧 −2 = 𝑝−2
21) Prove that the lines
𝑥+1
3
=
𝑦+3
5
=
𝑧+5
7
and
𝑥−2
1
=
𝑦−4
4
=
𝑧−6
7
are coplanar . Also, find
the plane containing these two lines.
𝑥
𝑦−1
1
2
22) Find the image of the two point ( 1, 6, 3) in the line =
=
𝑧−2
3
Prepared by Mr. Anirban Nayak
PGT, Mathematics, DPS Jodhpur
Mobile No.- 9828353006
23) Find the distance of the point ( 1, -2, 3) from the plane 𝑥 − 𝑦 + 𝑧 = 5 measured
𝑥
𝑦
2
3
along a line parallel to =
=
𝑧
−6
24) Find the shortest distance and the vector equation of the line of shortest distance
between the lines given by 𝑟⃗ = 3𝑖̂ + 8𝑗̂ + 3𝑘̂ + 𝜆(3𝑖̂ − 𝑗̂ + 𝑘̂ ) and 𝑟⃗ = −3𝑖̂ − 7𝑗̂ +
6𝑘̂ + µ(−3𝑖̂ + 2𝑗̂ + 4𝑘̂ )
25) Find the equation of the shortest distance line between the lines
and
𝑥+1
1
=
𝑦−4
1
=
𝑥+4
3
=
𝑦−1
5
=
𝑧+3
4
𝑧−5
2
Prepared by Mr. Anirban Nayak
PGT, Mathematics, DPS Jodhpur
Mobile No.- 9828353006