# Types of vectors and algebraic operations

Equal vectors: Vectors with the same length and direction, and must represent the same quantity (such as force or velocity).

Unit vector: For this vector, the length is always 1.For a vector 𝑎 ⃗ , a unit vector is in the

the same direction as 𝑎 ⃗ and is given by:

Zero vector: When the initial and terminal points of the vector coincide, we obtain a zero vector or null vector and it is denoted as . For example, or represents a zero vector.

Parallel vectors:

Coinitial vectors: Two or more vectors that have the same initial points are called coinitial vectors. Eg.and are coinitial vectors.

Collinear vectors: . . This means that they have one common point and the same direction.

Example: Show that P(0, 2, 4), Q(10, 0, 0) and R(5, 1, 2) are collinear.

have a common direction and a common point. Therefore P, Q and R are collinear.

Negative of a vector:Negative of a vector or Inverse vectors have the same length, but opposite direction.

Vector Addition:The sum of two or more vectors = resultant vector.The resultant vector can replace the vectors from which it is obtained.It is completely cancelled out by adding it to its inverse, which is then called the equilibrant.

## Triangle Law:

Let represent a vector a and represent a vector b. Then represents the vector c in the triangle shown below:

## Parallelogram law:

Arrange the tail of the vector to tail in the correct direction and draw to scale.Now sketch two identical vectors as the originals to form a parallelogram.Draw in the diagonal of the parallelogram. This is your answer called a resultant.Measure the resultant and find the angle.

Vector multiplication with scalar: Let and be two vectors and g and h as the scalar quantity. Then we can assume that:

## Properties of vectors:

1. Commutative Property: A+B = B+A

2. Associative Property: (A+B)+C = A+(B+C)

3. Zero Property: A+(-B) = 0, iff, A has the same magnitude to B and pointing in the opposite direction.

4. Subtraction: A – B = A + (-B)

5. Multiplication: 3 x A = 3A

Components of a vector: In three dimensions, the vector components of vector A are three perpendicular vectors AxAyand Az that are parallel to the x, y and z axes, respectively, and add together vectorially so that

A = Ax + Ay + Az

In this case: Ax = A cos Qx, Ay = A Sin Qy and Az = A Tan Qz

The position vector of a point that divides a line segment in a given ratio:

Let the point on line segment AB :A = (2, 7, 8) B = ( 2, 3, 12)

X divides [AB]in the ratio means

Example: P divides [AB] internally in ratio 1:3. Find P

point P is (2, 6, 9)

Example: X divide [AB] externally in ratio 2:1, or X divide [AB] in ratio –2:1. Find Q

point Q is (2,– 1,16)