Review: Analysis vector

VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR 1.5SCALAR AND VECTOR PRODUCT OF VECTORS

A scalar quantity – has only magnitude A vector quantity – has both magnitude and direction 1.1SCALARS & VECTORS electric field intensity

A vector in Cartesian Coordinates maybe represented as 1.2VECTOR COMPONENTS & UNIT VECTOR Or

The vector has three component vectors, which are, and VECTOR COMPONENTS & UNIT VECTOR (Cont’d)

Each component vectors have magnitude which depend on the given vector and they have a known and constant direction. A unit vector along is defined as a vector whose magnitude is unity and directed along the coordinate axes in the direction of the increasing coordinate values VECTOR COMPONENTS & UNIT VECTOR (Cont’d)

Any vector maybe described as The magnitude of written or simply given by VECTOR COMPONENTS & UNIT VECTOR (Cont’d)

Unit vector in the direction of the vector is: VECTOR COMPONENTS & UNIT VECTOR (Cont’d)

EXAMPLE 1 Specify the unit vector extending from the origin toward the point

SOLUTION TO EXAMPLE 1  Construct the vector extending from origin to point G  Find the magnitude of

 So, unit vector is: SOLUTION TO EXAMPLE 1 (Cont’d)

1.3VECTOR ALGEBRA Two vectors, and can be added together to give another vector Let

VECTOR ALGEBRA (Cont’d) Vectors in 2 components

Vector subtraction is similarly carried out as: VECTOR ALGEBRA (Cont’d)

Laws of Vectors:  Associative Law  Commutative Law  Distributive Law  Multiplication by Scalar

EXAMPLE 2 If Find: (a)The component of along (b)The magnitude of (c)A unit vector along

(a)The component of along is (b) SOLUTION TO EXAMPLE 2

Hence, the magnitude of is: (c) Let SOLUTION TO EXAMPLE 2 (Cont’d)

So, the unit vector along is: SOLUTION TO EXAMPLE 2 (Cont’d)

A point P in Cartesian coordinate maybe represented as The position vector (radius vector) of point P is as the directed distance from the origin O to point P is 1.4 POSITION AND DISTANCE VECTOR

POSITION AND DISTANCE VECTOR (Cont’d)

If we have two position vectors, and, the third vector or “ distance vector” can be defined as:

Point P and Q are located at and. Calculate: (a) The position vector P (b) The distance vector from P to Q (c) The distance between P and Q (d) A vector parallel to with magnitude of 10 EXAMPLE 3

(a) (b) (c) Since is the distance vector, the distance between P and Q is the magnitude of this distance vector. SOLUTION TO EXAMPLE 3

SOLUTION TO EXAMPLE 3 (Cont’d) Distance, d (d) Let the required vector be then Where is the magnitude of

Since is parallel to, it must have same unit vector as or SOLUTION TO EXAMPLE 3 (Cont’d) So,

Enclosed Angle SCALAR PRODUCT OF VECTORS 1.5 SCALAR AND VECTOR PRODUCT OF VECTORS

Surface and VECTOR PRODUCT OF VECTORS

Add the first two Columns Sarrus Law [Pierre Frédéric Sarrus, 1831] VECTOR PRODUCT OF VECTORS (Cont’d)

 Properties of cross product of unit vectors: Or by using cyclic permutation: VECTOR PRODUCT OF VECTORS (Cont’d)

Determine the dot product and cross product of the following vectors: EXAMPLE 4

The dot product is: SOLUTION TO EXAMPLE 4

The cross product is: SOLUTION TO EXAMPLE 4 (Cont’d)