Chapter 3( 3 d space vectors)

BMM 104: ENGINEERING MATHEMATICS I of 22

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


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


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


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;


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 .


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 .


Example:

Important:

(a)

(b)

(c)

(d)

.

If then .

Note:


(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


=

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 .


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


2. Direction of follows right-handed screw turned from to

3. Modulus of is Remarks:

Modulus of is therefore

or


Unit Vector :

4.

and

Similarly, and

and

Determinant Formula for


=

=

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.


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


By the sine rule:

Area of

=

=

Example: Find area for a triangle with vertices , and .

Equations (Vector, parametric and Cartesian equations) of a line


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.


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 .


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


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 .


(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 .


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)


5. 39 Joule

6. (a)

7. (a) (b) ,

8. (a)

(b) ,

(c)

9. (a) (b)

10. (a) (b)

11. (a) (b)

Chapter 3( 3 d space vectors)

Education

Transcript of Chapter 3( 3 d space vectors)