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Three Dimensional Co-ordinate Geometry Advanced Level Pure MathematicsChapter 7 Vectors
7.8 Vector Equation of a Straight Line 2
Chapter 10 Three Dimensional Coordinates Geometry
10.1 Basic Formulas 5
10.2 Equations of Straight Lines 5
10.3 Plane and Equation of a Plane 11
10.4 Coplanar Lines and Skew Lines 22
7.8 Vector Equation of a Straight Line
,
Remark
Example Let and .
(a) Find the equation of the straight line .
(b) Find the perpendicular distance from the point to the line .
Find also the foot of perpendicular.
Remark In above example (b), the distance from to may also be found directly without
calculating the foot of perpendicular. The method is outlined as follows:
By referring to Figure,
Since
Example By finding the foot of perpendicular from the point to the line,
, find the equation of straight line passing through and perpendicular
to , find the perpendicular distance from to .
Three Dimensional Co-ordinate Geometry
10.1 Basic Formula
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics
The Distance Between Two Points
Distance between and is .
Section Formula
Let divide the joint of and in the ratio
The Coordinate of the point is
10.2 Equations of Straight LinesIn vector form, the equation of straight line is , where is the position vector of any point in the
line, is fixed point on line and is direction vector of line.
If , , , we have
==
Since are basis vectors in , we have
or
Parametric Form of a Straight LineThe equation of the straight line passing through the point and with direction vector can
be expressed in the form of where is a parameter.
This is called the parametric form of the straight line.
Symmetric Form of a Straight LineThe equation of the straight line passing through the point and with direction vector and
is
and this is called the symmetric form of the straight line.
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Three Dimensional Co-ordinate Geometry Advanced Level Pure MathematicsGeneral Form of a Straight LineThe equation of a straight line can be written as a linear system
which is called the general form of a straight line.
If given two points , , the equation of straight line becomes
or
Example Find the equation of the line joining the points and .
S 1
Let and
To find the intersection point of line
we solve
i.e. find .
Note After finding is any two equations, must put into the 3rd equation in
order to test whether it is satisfied or not.
S 2
Distance of a point from the line
FIND .
Let be .
Direction vector of
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Three Dimensional Co-ordinate Geometry Advanced Level Pure MathematicsDirection vector of line
As is formed, can be determined and so
Theorem Given and
Their direction vectors are parallel
Remark
10.3 Plane and Equation of PlaneA vector perpendicular to (or orthogonal to) a plane is a normal vector o that
plane. In Figure, is a normal vector of the plane .
Normal vector of a plane is not unique, for if is a normal vector, then (a is
any non-zero real number) is also a normal vector.
Let be a fixed point and be any point on it.
Set i.e. A, B, C are given.
( Vector Form )
We have
( Normal Form )
Remark The general form of plane equation is .
Furthermore, if three points are given, .
We have
The system has non-trivial solution of .
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics
Hence, . It is an equation of plane. ( 3 Point Form )
Example Find the equation of the plane passing through the points , and .
Find also its distance from the origin.
The perpendicular distance between a point and a plane
Theorem The perpendicular distance between a point and a plane is
Proof Let be any point on the plane . is a vector normal to the plane .
The unit vector normal to the plane is .
The perpendicular distance between the point and the plane is equal to the magnitude of the projection of on . Therefore =
=
=
=
But, , since lies on the plane.
Example Find the perpendicular distance between two parallel planes and .
Solution Take a point on .The required distance is just the perpendicular distance between and .
i.e. = = units.
Angles Between Two planesGiven 2 planes and
The angle between two planes is and , which are a pair of supplementary angles and
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics
=
=
Remark (a)
(b)
Equation of Plane Containing Two Given Lines
Given two lines
The normal vector of the required plane
=
=
=
=
The equation of the plane
Example Find the equation of the plane containing two intersecting lines.
and
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics
Example
Solve
Solution
From the above examples we conclude that the
intersection of two planes is a line.
Alternatively,
consider
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics
Family of PlanesGiven two planes
The family of planes is any plane containing the line of intersection .
, where k is a constant.
Example Find the equation of the plane containing the line and passing
the point .
Example Find the equation of the plane containing the line and parallel to
the line .
Example (a) The position vector of a point is given by .
In Figure, is a point on the plane .
The line where is a real scalar and passing
through and does not lie on .
Show that the projection of on is given by where is a real
scalar.
(b) Consider the lines
and
and the plane
(i) Let and be the points at which intersects and respectively.
Find the coordinates of and and show that is perpendicular to both
and .
(ii) Show that the projections of and on are parallel.
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics
Theorem Two given planes and .
Prove that the equation of any plane through the line of intersection of must
contain a line
Proof The equation of plane through the line of intersection of is
Normal Vector of (*) .
Direction vector of line
is parallel to line .
Since and pass through the point .
contains .
10.4 Coplanar Lines and Skew Lines
Coplanar Lines
Definition Two lines are said to be Coplanar if there exists a plane that contains both lines.
Two lines are Coplanar they must be either parallel or they intersect.
Theorem Two lines and
are coplanar if and only if
Example Show that the two lines
and
are coplanar.
Skew Lines
Two straight lines are said to be Skew if they are non-coplanar i.e. neither do they intersect nor are they
being parallel.
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics
To find the shortest distance between them, we have to find the common perpendicular to both lines
first. The method is illustrated by the following example.
Example It is given that the two lines
and
are non-coplanar. Find the shortest distance between them.
Example Consider the line and the plane .
(a) Find the coordinates of the point where intersects .
(b) Find the angle between and .
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