Download THREE DIMENSIONAL GEOMETRY KEY POINTS TO REMEMBER

THREE DIMENSIONAL GEOMETRY
KEY POINTS TO REMEMBER
Line in space
 Vector equation of a line through a given point with position vector
vector
is
.
 Vector equation of line through two points with position vectors ,
=
and parallel to the
is
)
 Angle θ between lines
is given by cosθ =
and
 Two lines are perpendicular to each other if
= 0.
Plane
 Equation of a plane at a distance of d unit from origin and perpendicular to
 Equation of plane passing through
and normal to
is
.
is
= d.
= 0.
 Equation of plane passing through three non collinear points with position vectors
is (
= 0.
 If three points are collinear, there are infinitely many possible planes passing through
them.
 Planes passing through through the intersection of planes
= and
=d2 is
given by (
 Angle
)+λ(
) = 0.
between the two planes
=
=d2 is given by cos θ =
and
 Two planes are perpendicular to each other iff
 Two planes are parallel iff
 A line
=λ
= 0.
for some scalar λ
is parallel to the plane
0.
= d iff
ASSIGNMENT
1. Show that the four points (0, -1, -1), (-4, 4, 4), (4, 5, 1) and (3, 9, 4) are coplanar . Find the
equation of plane containing them.
2. Find the equation of plane passing through the points (2, 3, 4), (-3, 5, 1) and (4, -1, 2).
3. Find the equation of the plane through the points (2, 1, -1) and (-1, 3, 4) and perpendicular
to the plane x - 2y + 4z =10.
4. Show that the following planes are at right angles:
. (2
) = 5 and . (
)=3
5. Determine the value of p for which the following planes are perpendicular to each other:
i. . (
) = 7 and . (
) = 26
.
ii. 2x - 4y + 3z = 5 and x + 2y + pz = 5.
6. Find the equation of the plane passing through the point (-1, -1, 2) and perpendicular to
the planes 3x + 2y - 3z = 1 and 5x - 4y + z = 5.
7. Obtain the equation of the plane passing through the point (1, -3, -2) and perpendicular to
the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.
8. Find the equation of the plane passing through the origin and perpendicular to each of
the planes x + 2y - z = 1 and 3x - 4y + z = 5.
9. Find the equation of the plane passing through the points (1. -1, 2) and (2, -2, 2) and which
is perpendicular to the plane 6x - 2y + 2z = 9.
10. Find the equation of the plane through the point (1, 4, -2) and parallel to the plane
-2x + y -3z = 7.
11. Find the vector equations of the planes through the intersection of the planes
. (2
)+ 12 = 0 and . (
) = 0 which are at unit distance from the origin.
12.If the line
. (3
)+λ(
=(
) is parallel to the plane
) =14, find the value of m.
13. Find the equation of the plane through the points (1, 0, -1) ,(3, 2, 2)and parallel to the line
=
14. Find the equation of the plane passing through the point (0, 7, -7) &
containing the line
=
15. Prove that the lines
=
and
=
are coplanar. Also,
find the plane containing these two lines.
16. Show that the lines given below are coplanar. Also find the equation of
the plane containing them.
)+λ(
=(
) and
=(2
)+
(
)
17. Find the image of the point (3, -2, 1) in the plane 3x - y + 4z = 2.
18. Find the length and the foot of perpendicular from the point (7, 14, 5) to the plane
2x + 4y - z = 2.
19. Find the reflection of the point (1, 2 -1) in the plane 3x - 5y + 4z = 5.
20. Find the coordinate of the foot of perpendicular drawn from the point (5, 4, 2) to the line
=
. Hence or otherwise deduce the length of the perpendicular.
21. Find the image of the point with position vector 3
. (2
in the plane
) = 4.
22. Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane
.(
+ 4 ) + 5 = 0.
=
23. The cartesian equation of a line AB is
. Find the direction cosines of
a line parallel to AB.
=
24. Show that the lines
and
=
intersect. Also find the
point of intersection.
25. Find the coordinates of the point where the line through the points (3, -4, -5) and (2, -3, 1)
crosses the plane 2x + y + z = 7.
26. Find the vector and cartesian equation s of the planes containing the two lines
=( 2
)+λ(
) and =( 3
)+ (
).
27. Find the equation of the plane passing through the point (1, 1, 1) and containing the line
=( -3
) + λ (3
). Also show that the plane contains the line
)+λ(
=( -
).
28. Two airships are moving in space along the lines
=
and
=
An astronaut wants to move from one ship to another ship when the two airships are closest.
What is the least distance between the airships?
29. A bird is located at A (3, 2, 8). She wants to move to the plane 3x + 2y + 6z + 16 = 0 in
shortest time. Find the distance she covered.
30. By computing the shortest distance determine whether the following pairs of lines intersect
or not:
=
and
=
.
31. From a point A (2, 3, 8) in space , a shooter aims to hit the target at P(6, 5, 11). If the line of
fire is
=
, what you think about the success of the shooter?
32. An astronaut at A(7, 14, 5) in space wants to reach a point P on the plane 2x + 4y - z = 2
when AP is least. Find the position of P and also the distance AP travelled by the astronaut.
33. Find the distance of the point (3, 4, 5) from the plane x + y + z = 2 measured parallel to the
line 2x = y = z.
Answer key
1. 5x - 7y + 11z +4 = 0
6. 5x + 9y + 11z - 8 =0
10. 2x - y + 3z + 8 = 0
12. m = -2
16. . (
2. x + y -z = 1
3. 18x + 17y + 4z = 49 5. i. 17 ii. 2
7. 2x - 4y + 3z -8 = 0
8. x + 2y + 5z = 0 9. x + y - 2z + 4 = 0
11. . (2
+ 2 )+ 3 = 0 and . (
)+3=0
13. 4x - y - 2z - 6 = 0
14. x + y + z = 0
15. x - 2y + z = 0
)+7=0
17. (0, -1, -3)
18. (1, 2, 8); 3
units
19. (73/25, -6/5, 39/25)
20. (1, 6, 0) ; 2
units
21. (1, 2, 1)
22.
; (-1/12, 25/12, -2/12)
26.
.(
28. 2
,
23.
,
24. (-1, -1, -1)
) - 37 = 0; 10x + 5y -4z - 37 = 0
units
29. 11 units
31. he will be successful
25. (1, -2, 7)
27. x - 2y + z = 0
30. lines do not intersect
32. P(1, 2, 8); 3
units
33. 6 units
______________________________________________________________________________
VECTOR ALGEBRA
IMPORTANT POINTS TO REMEMBER
 A quantity that has magnitude as well as direction is called a vector. It is denoted by a
directed line segment.
 Two or more vectors which are parallel to same line are called collinear vectors.
 Position vector of a point P(a, b, c) w.r.t. origin (0, 0, 0) is denoted by
, where
=
a
and l
l=
.
 If A(x1, y1, z1) and B (x2, y2, z2) be any two points in space, then
= (x2 - x1) + (y2 - y1) + (z2 - z1) and
l
l=
 If two vectors
and
are represented in magnitude and direction by the two sides of a
triangle taken in order, then their sum
is represented in magnitude and direction by
third side of triangle taken in opposite order. This is called triangle law of addition of
vectors.
 If is any vector and is a scaler, then λ is a vector collinear with and lλ l = lλl l l.
 If
and
are two collinear vectors, then
 Any vector
 If
and
can be written as
= λ , where λ is some scaler.
= l l , where
is a unit vector in the direction of .
be the position vectors of points A and B, and C is any point which divides
in the ratio m : n internally then position vector
divides
 The angles
in ratio m : n externally, then
made by
=
of point C is given as
=
. If C
.
= a +b + c with positive direction of x, y and z-axis
are called direction angles and cosines of these angles are called direction cosines of
denoted as l = cos , m = cos , n = cos . Also l =
,m=
,n=
 The numbers a, b, c proportional to l, m, n are called direction ratios.
& l 2 + m 2 + n2 = 1
 Scaler product of two vectors
lcosθ where

and
is the angle between
is denoted as .
and . (0
if and only if

= l l2, so
 If
= a1 + a2 +a3 ,
= l ll
).
= 1.
= b1 + b2 +b3 , then
= l ll l sinθ
x
.
is perpendicular to
 Cross product ( Vector product ) of two vectors
as
& defined as
,where
a1b1 + a2b2 + a3b3.
and
is the angle between
unit vector perpendicular to both
and
such that ,
is denoted as x
and . (0
and
& defined
) and
is a
form a right handed
system.

x
=
iff
is parallel to .
 Unit vector perpendicular to both & =
 l x
l is the area of parallelogram whose adjacent sides are

is the area of parallelogram whose diagonals are
and
and
ASSIGNMENT
1. If
are the position vectors of the points (1, -1), (-2, m), find the value of m for which
are collinear.
2. If a vector makes angles
+ sin2
with OX, OY and OZ respectively, prove that
= 2.
3. Find the direction cosines of a vector
l
which is equally inclined with OX, OY and OZ. If
l is given, find the total number of such vectors.
4. A vector
is inclined at equal angles to OX, OY and OZ. If the magnitude of
is 6 units,
find
5. A vector
has length 21 and d. r.s 2, -3, 6. Find the direction cosines and components
of , given that it makes an acute angle with x axis.
6. Find the angles at which the vector 2 - + 2 is inclined to each of the coordinate axes.
7. For any vector , prove that
=( . ) +( . ) +( . ) .
8. Find the value of p for which the vectors
i. Perpendicular
9. If
10. If
3 +2 +9 and
+ p +3 are
ii. Parallel
are unit vectors inclined at an angle , then prove that sin = l
makes equal angles with
and each of
&
is cos-1(1/
&
l.
and has magnitude 3, then prove that the angle between
).
11. If
+
+ = , l l = 3, l l = 5 and l l = 7, find the angle between
.
12. Find a vector of magnitude 9, which is perpendicular to both the vectors 4 - +3 &
-2 + -2 .
13. Find a unit vector perpendicular to the plane ABC where A, B, C are the points (3, -1, 2),
(1, -1, -3), (4, -3, 1 ) respectively.
14. Show that area of a parallelogram having diagonals 3 + - 2 and -3 + 4 is 5 .
15. Show that ( x )2 =
16. Prove that the points A, B & C with position vectors
and only if
17. If
x
x
= x
18. Let
+
and
x
x
=
= .
x
show that
be unit vectors such that
prove that
19. If
+ x
respectively are collinear if
.
=
is parallel to
.
, where
= 0 and the angle between
is ,
x
are three vectors such that
+
+ = , then prove that
x
=
x
= x .
20. If
are vectors such that . = . , x = x ,
, then show that
21. Show that the vectors 2 -3 + 4 and -4 +6 -8 are collinear.
22. Find , if (2 +6 + 14 ) x ( -λ + 7 ) = .
23. Let
+ 4 +2 ,
- 2 +7 and
is perpendicular to both
24. If
≠
,
and .
are three vectors such that l
2 - +4 . Find a vector
.
which is
= 18.
l = 5, l
l = 12 and l
l = 13, and
+
+ = ,
find the value of . + . + . .
25. Find the position vector of a point R which divides the line joining the two points P and Q
whose position vectors are (2 +
and (
) respectively, externally in the ratio 1 : 2.
Also, show that P is the midpoint of the line segment RQ.
26. Find a unit vector perpendicular to each of the vectors
+ +2 ,
+2 -2 .
27. If l
l = 13, l
28. If l
l = 2, l
29. If
l = 5 and
.
= 60, then find l x
l = 5 and l
x
l = 8, find
+ + ,
x = and . = 3.
30. Express the vector
l.
. .
are given vectors, then find a vector
satisfying the equations
- 2 + 5 as the sum of two vectors such that one is parallel to
the vector
+ and other is perpendicular to .
31. If the vertices A, B, C of triangle ABC have position vectors (1, 2, 3) (-1, 0, 0), (0, 1, 2)
respectively, what is the magnitude of angle ABC?
32. If is a unit vector, then find l l such that (
). (
) = 8.
33. Show that vector + + is equally inclined to the axes.
34. Show that three points
,
and
are collinear.
35. If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is
.
Answer Key
1. m = 2
3. ±1/ , ±1/
±1/ ; 8 ways
5. 2/7, -3/7, 6/7 ; = 6 -9 + 18
6. =
8. i. p = -15 ii. p = 2/3
22. -3
23.
(64 - 2 - 28 )
27. 25
31. cos-1 (
11.
28. 6
)
12. -3 + 6 + 6
24. -169
29.
32. 3
4.
25. 3
=2
=
(± ± ± )
=
13.
(-10 -7 +4 )
(
-
-
30. 6 + 2 , - -2 + 3