# Vector Analysis

Review of Plane Trigonometry

Definitions and Laws of Vector Algebra:

1. A unit vector is defined as a vector whose magnitude is unity. If we divide a vector by its magnitude, we obtain a unit vector in the direction of the original vector. A unit vector can be expressed as

 (3)

We can also express any vector in terms of its magnitude and the unit vector in the same direction as

 (4)

2. A vector may be represented in rectangular Cartesian coordinates as

 (5)

where are the unit vectors along x, y, z axes, respectively.

3. The magnitude of a vector can be determined as

 (6)

4. Two vectors are equal only if they have the same magnitude and direction. This condition can be described mathematically as follows:

Vector is equal to vector only when

5. When two or more vectors are added together, the resulting vector is called the resultant.

6. If a vector is multiplied by a scalar as in , then the magnitude of the resulting vector is equal to the product of p and the magnitude of , and its direction is the same as if p is positive and opposite to if p is negative.

6.1 Associative law for scalar multiplication:

6.2 Distributive law for scalar multiplication:

7. A vector can be multiplied by another vector either through a dotor a crossproduct

7.1 Dot product of two vectors results in a scalar quantity as shown below

, where q is the angle between vectors and .

The dot product is performed as

In dot product, the order of the two vectors does not change the result.

The associative law of multiplication also applies to the dot product.

7.2 Cross product of two vectors results in another vector quantity as shown below

, where and q is the angle between vectors and . The direction of vector is perpendicular to the plane containing vectors and such that follow the right hand rule.

 The cross product is performed as In cross product, the order of vectors is important. The associative law of multiplication also applies to cross product.

Review of Plane Trigonometry