chapter 3( 3 d space vectors)
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for engineering math 1TRANSCRIPT
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CHAPTER 3: 3-D SPACE VECTORS
Basic Concept of Vector
A vector is a quantity that having a magnitude/length (absolute) and a direction.
Notation: i. or is vector
ii. or is modulus/absolute value/length of the vector
is opposite to
Meanwhile = .
Characteristics of vectors
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1. If is parallel to , then OR where k and t are the scalars or parameters.
2. If OR , then and are in the opposite directions.
3. If OR , then they are in the same direction.
Example:
i) or
ii) or
Addition Law of Vectors
Let , . Refer to below diagram.
If is the position vector for A and is the position vector for B then
BUT
Example: Find Components of Vectors
2-D Space
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2 basic vectors : and
They are also called unit vectors as and
Position vector of point is given by .
Absolute/ modulus by Pythagoras Theorem
Example:
i)
3-D Space
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o All the x-axes, y-axes are perpendicular to each other.
o There are 3 basic vectors: , , .
o They are all unit vectors that parallel to the axes respectively and thus they also perpendicular to each other.
Position vector of is and
Length is
Example: Given . Find
Addition of vectors for 3-D Space
The sum of two vector and is the vector formed by
adding the respective component;
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Subtraction of vectors for 3-D Space
The subtraction of two vectors and is the vector formed
by adding the respective component;
Example: Given , and . Find
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(v)
Unit Vector in the Direction of
is a unit vector in the direction of .
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Example: Given . Find unit vector in the direction of ?
Dot/ Scalar Product
Notation for Dot product:
The dot product of and is the real number obtained by
where is the angle between and and .
When we measuring angle between two vectors, the vectors must have the same initial point.
Example: (i) Given and . Find .
(ii) Given and and . Find .
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Example:
Important:
(a)
(b)
(c)
(d)
.
If then .
Note:
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(i) If then
(ii)
Properties of the dot product
a) : Cumulative
b) : Distributive
c)
Example:
Finding angle between the vectors and given that .
Formula to compute scalar product
Example: Given and . Verify = .
Remark: By using addition law of vectors and the law of cosine, it can be showed that
.
Component of in the direction of : Denoted as
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=
OQ is called component of vector in the direction of
= =
= ^
~ncosa
since
By the definition of product
Example:
Find
(i) given and .
(ii) given and .
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Application of Dot Product : WORK DONE
Work done = Magnitude of force in the direction of motion times the distance it travels =
Example: A force causes a body to move from to . Find
the work done by the force.
Vector/ Cross Product
Definition: is defined as a vector that
1. is perpendicular to both and .
and
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2. Direction of follows right-handed screw turned from to
3. Modulus of is Remarks:
Modulus of is therefore
or
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Unit Vector :
4.
and
Similarly, and
and
Determinant Formula for
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=
=
Example: Find , and verify that given and .
Applications of Cross Product
1. The moment of a force
A force is applied at a point with position vector to an object causing the object to rotate around a fixed axis.
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As the magnitude of moment of the force at O is
= ( Magnitude of force perpendicular to d ) (Magnitude of displacement)
Thus we have
Therefore we define the moment of the force about O as the vector
As
The magnitude, , is a measure of the turning effect of the force in unit of Nm.
Example: Calculate the moment about O of the force that is applied at the point
with position vector 3j. Then calculate its magnitude.
2. Calculate the area of a triangle
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By the sine rule:
Area of
=
=
Example: Find area for a triangle with vertices , and .
Equations (Vector, parametric and Cartesian equations) of a line
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Let and as a vector that parallel to the line L.
As thus , is a parameter (scalar).
By the addition law of vectors, we obtain
i.e. the vector equation of a line passing through a fixed point and parallel to a vector
is .
are the parametric equations of L.
is the Cartesian equation of L.
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Example: Find the vector equation of line passes through and .
Example: Find Cartesian equation of line passes through and .
Equation (Vector and Cartesian equations) of a plane
Vector equation:
Cartesian equation:
Remark: In general, OR is the Cartesian equation of a
plane with a normal vector .
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Example: A plane contains , and . Find the vector and Cartesian forms of the equation of the plane.
and
Distance From A Point to A Line and to A Plane
Distance From A Point to A Line
Distance, d, from a fixed point P to a line: where A is a point on the line
and is a vector parallel to the line is given by
Example: Find the distance from point to the line .
Distance From A Point to A Plane
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Distance, D, from a fixed point P to a plane where A is a point on the plane
and is a normal vector to the plane is given by
.
Example: Find the distance from point to the plane .
PROBLEM SET: 3-D SPACE VECTORS
1. Points , and have coordinates , and respectively. Find
(a) the position vectors of P, Q and R.
(b) and
QR .
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(c)
PQ and .
2. A triangle has vertices , and respectively. Calculate the vectors which represent the sides of the triangle.
3. Find and verify that if
(a) , (b) ,
4. (a) Find the component of the vector in the direction of the vector
.
(b) Find the component of the vector in the direction of the vector
.
5. A force causes a body to move from point to point .
Find the work done by the force.
6. (a) If and , find .
(b) Verify that .
7. (a) Find the area of the triangle with vertices , and .
(b) A force of magnitude 2 units acts in the same direction of the vector
. It causes a body to move from point to point
. Find the work done by the force.
8. Find the vector equation of the line passing through(a) and .
(b) the points with position vectors and .
Find also the cartesian equation of this line.
(c) and which is parallel to the vector .
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9. Given , and . Find(a) the area of the triangle ABC.(b) the Cartesian equation of the plane containing A, B and C.
10. (a) Find the distance from point to the line .
(b) Find the distance from point to the plane .
11. (a) Find the distance from to the plane
(b) Find the distance from point to the line .
ANSWERS FOR PROBLEM SET: 3-D SPACE VECTORS
1. (a) , ,
(b) (c)
2. , ,
3. (a) (b) 22
4. (a) (b)
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5. 39 Joule
6. (a)
7. (a) (b) ,
8. (a)
(b) ,
(c)
9. (a) (b)
10. (a) (b)
11. (a) (b)