calculus-iii by dr. umer farooq comsats institute of information technology (ciit), islamabad...

Calculus-III

by Dr. Umer Farooq

Comsats Institute of Information Technology (CIIT), Islamabad

Lecture No 3

Outlines of Lecture 02

•Vector Addition, Subtraction and Scalar Multiplication•Unit Vector, Normalization of a Vector•Vectors Determined by Length and Angle•Resultant of Two and Three Forces

Dot Product and Projections

Definition: If and are vectors in 2-space then the

dot product between and is defined as and is defined as Similarly, if and are

vectors in 3-Space then the dot product is defined as

),( 21 uuu

),( 21 vvv

u

v

vu

2211 vuvuvu

),,( 321 uuuu

),,( 321 vvvv

332211 vuvuvuvu

Dot Product and Projections

Example 3.1 Find the dot product in the following cases.

Solution:

Dot Product and Projections

vIf , and are three vectors , then the following results hold

u

w

Dot Product and Projections

uTheorem: If and are vectors in 2-space or 3-space and if is the angle between them then

Proof:

v

Dot Product and Projections

Dot Product and Projections

Example 3.2 Find the angle between the vector and(a) (b) (c)

Solution:

kjiu22

kjiv263

kjiw672

kjiz663

Dot Product and Projections

Dot Product and Projections

Dot Product and Projections

Perpendicular and Parallel Vectors:

Dot Product and Projections

Direction Angles In an xy-coordinate system, the direction of a non-

zero vector is completely determined by the angles and between and the unit vectors and .

v

v i j

Fig.3.1

Dot Product and Projections

Fig.3.2

Direction Angles

Dot Product and Projections

Direction Cosines In both 2-space and 3-space the angle between a

non-zero vector and , and are called the direction angles of , and the cosines of those angles are called the direction cosines of .

If

v ij

k

v

v

Dot Product and Projections

Example 3.3 Find the direction cosines of the vector

and approximate the direction angles to the nearest degree.

Solution:

kjiv442

Dot Product and Projections

Dot Product and Projections

Example 3.4 Find the angle between a diagonal of the cube and one of its

edges.

Solution:

Decomposing Vectors into Orthogonal Vectors : In many applications it is desirable to “decompose” a

vector into a sum of two orthogonal vectors with convenient specified directions. For example fig. 3.3 shows an inclined plane. The downward force that gravity exerts on the block can be decomposed into the sum

Dot Product and Projections

F

21

FFF

Fig.3.3

Dot Product and Projections Decomposing Vectors into Orthogonal Vectors : where the force is parallel to the ramp and the

force is perpendicular to the ramp. The forces

and are useful because is the force that pulls the block along the ramp and is the force that block exerts against the ramp.

1

F2

F

1

F 2

F 1

F

2

F

Dot Product and Projections Decomposing Vectors into Orthogonal Vectors :

Dot Product and Projections Decomposing Vectors into Orthogonal Vectors :

Dot Product and Projections Decomposing Vectors into Orthogonal Vectors :

Dot Product and Projections

Example 3.5 Let , and

Find the scalar and vector components of along andSolution:

)3,2(

v )2

1,2

1(1 e )

2

1,2

1(2 e

v 1e

2e

Dot Product and Projections

Dot Product and Projections

Example 3.6 A rope is attached to a 100 lb block on a ramp that is inclined

at an angle of with the ground. How much force does the block exert against the ramp, and how much force must be applied to a rope in a direction parallel to the ramp to prevent the block from sliding down the ramp?

Solution:

30

Dot Product and Projections

Dot Product and Projections

Orthogonal Projections

Dot Product and Projections

Dot Product and Projections

Example 3.7Find the orthogonal projection of on and then find the vector component of orthogonal

to .Solution:

kjiv

jib22

v

b

Dot Product and Projections

Dot Product and Projections

Example 3.8 Find so that the vector from the point

to the point is orthogonal to the vector from to the point

Solution:

)3,1,1( A

)5,0,3(B

A ),,,( rrrP

r

Dot Product and Projections

Example 3.9 Find two unit vectors in 2-space that make an angle

of with . Solution:

Dot Product and Projections

45 ji34

Dot Product and Projections

Dot Product and Projections

Have a Nice Day Thank You

Vectors