Vector Addition

Cummutative Law

A

B

CB

A

C

A + B = C B + A = C

A + B = B + A

Vector AdditionAssociative Law

A

B

C

A + B

(A +

B) +

C

A

B

C

B +

C

AA

+ (B

+ C

)

Vector SubtractionSubtract B from A

A

B

-B

A+B

A-B

Right Hand RuleE, B, F

Right Hand Rule

E, B, F

Right Hand Rule

Right Hand Rule for “F”

Fingers in the direction of B. Thumb in the direction of I. Palm in the direction of F.

Right Hand Rule

Right hand rule for force:

Fingers in the direction of B. Thumb in the direction of I. Palm in the direction of F.

Right Hand Rule

What happens when a current is driven through the wire?

Force out of the screen. Force into the screen.

Resolution of Vector

Orthogonal Vectors

A

x

z

y

^

^ ^

ZY

YX

XZZYX

Resolution of Vector

Component Vectors

A = Ax + Ay + Az

A = xAx + yAy + zAz^ ^ ^

|A| =√Ax2 + Ay

2 + Az2

Vector Problems 1:

1. A = x6 + y3 B = x4 - y7^ ^ ^ ^

A + B = ?

Magnitude of Resulting Vector ?

Its angle with respect to x- axis ?

Solution of Problem 1

Unit Vectors x (6+4) =10

Unit Vectors y (3-7) = -4

Magnitude of Resultant Vector C=√102 + (-4)2

= √116 = 10.8

Angle α =tan-1 (-4/10) = -21.8º

Vector Problem 2:

Addition of Three Vectors

A = -x8 + y12; B = -x5 + y15; C = x7 – y9

Magnitude of Resultant Vector D ?

Angle of D with x-axis ?

Scalar Product

Product of their magnitudes

multiplied by the cosine of the

angle between the Vectors

Scalar / Dot Product of Two Vectors

Orthogonal Vectors

Angular Dependence

Scalar Product

Scalar Product of a Vector with itself ?

A . A = |A||A| cos 0º

= A2

Scalar Product

Scalar Product of a Vector and Unit vector ?

x . A =|x||A|cosα

= Ax

Yields the component of a vector in a direction of the unit

vector

Where alpha is an angle between A and unit vector x

^ ^

Scalar Product

Scalar Product of Rectangular Coordinate

Unit vectors?

x.y = y.z = z.x = ?

= 0

x.x = y.y = z.z = ?

= 1

Scalar Product Problem 3:

A . B = ?

( hint: both vectors have components in three directions of unit vectors)

Scalar Product Problem 4:A = y3 + z2;B= x5 + y8

A . B = ?

Scalar Product Problem 5:

A = -x7 + y12 +z3;

B = x4 + y2 + z16

A.B = ?

Cylindrical Coordinates

r =

=

z = z

Spherical Coordinates

=

=

=

x = r sin cos

y = r sin sin

z = r cos