Calculus-III by Dr. Umer Farooq Comsats Institute of Information Technology (CIIT), Islamabad Lecture No 3

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


Example 3.1 Find the dot product in the following cases.

Solution:


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

u

w


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

Proof:

v



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

Solution:

kjiu22

kjiv263

kjiw672

kjiz663




Perpendicular and Parallel Vectors:


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


Fig.3.2

Direction Angles


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


Example 3.3 Find the direction cosines of the vector

and approximate the direction angles to the nearest degree.

Solution:

kjiv442



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


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 :




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



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



Orthogonal Projections



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

to .Solution:

kjiv

jib22

v

b



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


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

of with . Solution:


45 ji34



Have a Nice Day Thank You

Vectors

Documents

Calculus-III by Dr. Umer Farooq Comsats Institute of Information Technology (CIIT), Islamabad Lecture No 3