M P3 CHAPTER 7
FM P1 CHAPTER 8
VECTORS
7.1 VECTOR EQUATION A LINE
- INTRODUCTION
Always use directional
vector if you are
applying
dot / scalar product or
cross / vector product
the position
vector of a fixed
point on the line
a variable
scalar
the direction
of / parallel to the
line
Note: There is no unique vector equations.
Dot / scalar product
to find angles
Cross / vector product
to find the common
perpendicular of 2
lines / normal vector
of a plane
IMPORTANT: If the dot product of
2 vectors is equal to zero, then the
2 vectors are perpendicular to
each other.
Plus Minus Plus
7.1 VECTOR EQUATION
- HOW THE QUESTION IS ASKED DURING THE EXAM
Example 7.1:
Find a vector equation that passes through the coordinate and is parallel to
the vector .
Answer:
Position vector
or
7.1 VECTOR EQUATION
- HOW THE QUESTION IS ASKED DURING THE EXAM
Example 7.2:
Find a vector equation for the line through the points and .
Answer:
or
7.2 EQUATION OF A PLANE
- INTRODUCTION
If
is the Cartesian equation of a plane, then
every
vector that is perpendicular to this
plane, i.e. the normal vector of this plane,
is
7.2 EQUATION OF A PLANE (FM)
- INTRODUCTION
Always use directional
vector if you are
applying
dot / scalar product or
cross / vector product
the position
vector of a fixed
point on the line
a variable
scalar
Two non-parallel
direction in
the plane
Note: There is no unique vector equations.
7.2 EQUATION OF A PLANE
- HOW THE QUESTION IS ASKED DURING THE EXAM
Example 7.3:
Find the Cartesian equation of the plane through the point with normal .
Solution:
The
equation is
The constant has to be chosen so that the plane passes through . The
constant is therefore
So the equation is
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7.2 EQUATION OF A PLANE
- HOW THE QUESTION IS ASKED DURING THE EXAM
Example 7.4:
Find the Cartesian equation of the plane , given , and .
Solution:
1.
Identify two vectors on the plane.
and
2.
Perform vector / cross product.
3.
Substitute any position vectors on the plane, say ,
4.
Now, we obtain the equation of the plane as
or
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7.2 EQUATION OF A PLANE
- HOW THE QUESTION IS ASKED DURING THE EXAM
Example 7.4:
Find the Cartesian equation of the plane , given , and .
Solution:
1.
Identify two vectors on the plane. and
Vector equation of a plane (for FM)
2. Perform vector / cross product.
3. Substitute any position vectors on the plane, say ,
Every time you perform cross product, you obtain
the
normal
vector the
of the
plane of the plane as
4.
Now,
we obtain
equation
Common perpendicular oforand
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7.3 PERPENDICULAR DISTANCE
- FROM A LINE INTRODUCTION
1. Find a point on the line such that
is perpendicular to .
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2. By using the dot product, solve
to obtain the value of .
3. By substituting this , you obtain
the coordinate of point . The
magnitude
of
is
the
perpendicular distance of from .
Foot of the
perpendicular
7.3 PERPENDICULAR DISTANCE
- FROM A LINE HOW THE QUESTION IS ASKED DURING THE EXAM
Example 7.5:
Find the perpendicular distance of the point from
the line .
Solution:
Let to be a point on the line so that is perpendicular
to the line.
.
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7.3 PERPENDICULAR DISTANCE
- FROM A PLANE INTRODUCTION
The perpendicular distance of a
point with position vector from the
plane is
Perpendicular
distance
Foot of the
perpendicular
7.3 PERPENDICULAR DISTANCE
- FROM A PLANE HOW THE QUESTION IS ASKED DURING THE EXAM
Example 7.6:
Find the distance of the point from the plane with equation
Solution:
7.3 PERPENDICULAR DISTANCE (FM)
- BETWEEN 2 LINES INTRODUCTION
Given two lines
then the shortest distance between two lines
is
The positive
value
The modulus
of
7.3 PERPENDICULAR DISTANCE (FM)
- BETWEEN 2 LINES HOW THE QUESTION IS ASKED DURING THE
EXAM
Example 7.7:
The position vectors of the points , , ,
are , , ,
respectively. Find the shortest distance between the line
and .
Solution:
7.4 INTERSECTION POINT
- BETWEEN 2 LINES HOW THE QUESTION IS ASKED DURING THE
EXAM
Example 7.8:
Find the value of for which the lines and intersect.
Solution:
Since
both lines intersect,
The first two equations give and .
Putting these values into the third equation gives .
7.4 INTERSECTION POINT
- BETWEEN A LINE & A PLANE
- HOW THE QUESTION IS ASKED DURING THE EXAM
Example 7.9: Find the coordinates of the point of intersection of the line
with the plane .
Answer:
Substitute into the equation of the plane,
7.4 INTERSECTION (A VECTOR EQN)
- BETWEEN 2 PLANES HOW THE QUESTION IS ASKED DURING THE
EXAM
Example 7.10:
Find the vector equation for the line of intersection of the planes
and
Solution:
By , we obtain
Let , then , and
Let , then , and
Therefore the vector equation for the intersection of the planes can be
or
Note that, the answers obtained are not unique.
7.5 ANGLE SCALAR PRODUCT
Between 2 lines or between 2 planes
Between a line and a plane
Note: acute angle
7.5 ANGLE SCALAR PRODUCT
Example 7.11:
Find the angle between the planes and .
Solution:
Normal vector of the plane will be used
Negative dot product at the RHS will not give acute angle.
7.5 ANGLE SCALAR PRODUCT
Example 7.12:
Find the acute angle between the line and the plane .
Solution:
Directional vector will be
used, not the position vector
Negative dot product at the RHS will not give acute angle.
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7.6 AREA OF A TRIANGLE (FM)
Area of a triangle
Example 7.13:
Find the area of triangle where the
given coordinates of is , is and is .
Solution:
Area ##
