CHENNAI PUBLIC

SCHOOL
Anna Nagar  Chennai -600 101

3D
SHORTEST DISTANCE BETWEEN THE SKEW LINES
1. Find the shortest distance between the lines whose vector equations are
r⃗ =( i^ + ^j ) + λ (2 i^ − ^j + k^ )and r⃗ =( 2 i+
^ ^j−k^ ) + μ (3 i−5
^ ^j+2 k^ )

2. Find the shortest distance between the lines whose vector equations are
^ ( 2 s−1 ) ^j+(2 s+1) k^ )
^ ( t−2 ) ^j+ ( 3−2 t ) ^k )and r⃗ =( s+1 ¿ i+
r⃗ =( 1−t ¿ i+
3. x−3 y−5 z−7 x+1 y +1 z +1
Find the shortest distance between the lines 1
=
−2
=
1
and 7
=
−6
=
1
4. x−1 y +1 x+1 y−2
Find the shortest distance between the lines 2 = 3 =z and 5 = 1 ; z=2
5. Find the shortest distance between the lines whose vector equations are
^ 2 ^j+ 2 k^ ) + λ ( i−2
r⃗ =( 6 i+ ^ ^j +2 k^ )and r⃗ =(−4 i−
^ k^ ) + μ (3 i−2
^ ^j−2 k^ )
6. Find the shortest distance between the line joining the points P(2,3,-4),Q(3,-1,5) and the line joining the
points A(-1,3,2),B(4,1,1).
ANGLE BETWEEN TWO LINES
1. −x+ 2 y−1 z +3 x+2 2 y−8 z−5
Find the angle between the lines −2
=
7
=
−3
and −1
=
4
=
4
2. ^
Find the angle between the lines r⃗ =( 2i−5 ^j+ k^ ) + λ(3 i+
^ 2 ^j+6 k^ )and r⃗ =( 2 i−6
^ k^ ) + μ( i+2
^ ^j+2 k^ ).
3. 1−x y −2 z−3 x−1 y −1 6−z
Find the value of λ so that the lines 3 = 2 λ = 2 and 3 λ = 1 = 7 are perpendicular to each
other.
4. x−1 y −2 z−3 x−1 y −2 z−3
If the lines −3 = −2 k = 2 and k = 1 = 5 are parallel find the value of k.
5. x−3 y−3 z
Find the equations of the lines through the origin which intersect the line 2
=
1
= at an angle of
1
π
.
3
EQUATION OF LINE THROUGH ONE POINT AND PERPENDICULAR TO TWO LINES
1. Find the vector equation of the line passing through the point (1,2,-4) and perpendicular to the two lines
x−8 y +19 z −10 x−15 y−29 z−5
= = and = =
3 −16 7 3 8 −5
2. Find the vector equation of the line passing through the point (-1,3,-2) and perpendicular to the two lines
x y z x+2 y−1 z +1
= = and = =
1 2 3 −3 2 5
EQUATION OF LINE THROUGH ONE POINT AND PARALLEL TO A LINE
1. Find the vector and cartesian equations of the line passing through the point (1,2,3) and parallel to the line
−x−2 y+ 3 2 z−6
= = .
1 7 3
2. Find the vector and cartesian equations of the line passing through the point (2,-1,1) and parallel to the line
joining the points (-1,4,1) and (1,2,2)
3. x−1 3− y z +1
Find the vector and cartesian equations of the line parallel to the line 5 = 2 = 4 and passing

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 through (3,0,-4).Also find the distance between these two lines.

PROBVLEMS BASED ON DISTANCE

1. Find the perpendicular distance of the point (1,0,0) from the line x−1 = y +1 = z+ 10
2 −3 8
2. x+2 y +1 z−3
Find the point on the line 3 = 2 = 2 at a distance 3 √ 2units from the point (1,2,3)
3. x+2 y +1 z−3
Find the point on the line 3 = 2 = 2 at a distance 5units from the point (1,3,3)
4. Write the vector equations of the following lines and determine the distance between them
x−1 y −2 z + 4 x−3 y−3 z+5
= = ; = =
2 3 6 4 6 12
5.
6. x−1 y −2 z−3 x−4 y +1
Show that the lines 2 = 3 = 4 ; 5 = 5 =z intersect and find the point of intersection
7. ^ ^j−k^ ) + λ ( 2 i+
Find whether the linesr⃗ =( i− ^ ^j ) and r⃗ =( 2i−
^ ^j ) + μ ( i+
^ ^j− k^ ) intersects find the point of intersection
8. x+5 y +3 z−6
Find the equation of the perpendicular drawn from the point P(2,4,-1) to the line 1 = 4 = −9
9. ^ ^j− k^ ) + λ ( 3 i−
Show that the linesr⃗ =( i+ ^ ^j ) and r⃗ =( 4 i−
^ k^ ) + μ ( 2 i+
^ 3 k^ ) intersect.Also find the point of
intersection.
10. x−1 y +2 z−3
Find the coordinates of the points on the line 2 = 2 = 6 which are at a distance of 3 units from the
point (1,-2,3).
11. Find the distance of the point (2,4,-1) from the point of intersection of the lines
x−4 y −5 z −1 x−3 y−9 z−4
= = ; = =
2 3 1 7 5 0
12. Find the coordinates of the foot of the perpendicular and the length of the perpendicular drawn from the
point P(5,4,2) to the line r⃗ =−⃗i +3 ⃗j+ ⃗k + λ ( 2 i⃗ +3 ⃗j− ⃗k ) .Also find the image of the point P in this line.
13. x y−1 z−2
Find the image of the point (1,6,3) in the line 1 = 2 = 3

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