Types of Vectors in Algebra

Vectors are quantities that have both magnitude and direction, while scalars have magnitude only. In Vector Algebra, understanding different types of vectors is important because these classifications are used in geometry, physics, and later topics such as direction ratios, dot product, and three-dimensional geometry.

1. Zero Vector (Null Vector):

A vector whose initial point and terminal point coincide is called a zero vector. Its magnitude is zero, and it does not have a definite direction in the usual sense.

• Notation: \[\vec{0}\]
• Key idea: No net displacement.

2. Unit Vector

A vector having magnitude 1 is called a unit vector. If a vector \[\vec{a}\] is given, then the unit vector in its direction is written as \[\hat{a}\].

• Key idea: It shows direction only.
• Common examples: \[\hat{i}, \hat{j}, \hat{k}\] along x, y, z axes.

3. Coinitial Vectors

Vectors having the same initial point are called coinitial vectors. Their magnitudes and directions may be different, but they begin from a common point.

4. Collinear Vectors

Vectors that are parallel to the same line are called collinear vectors. They may act in the same direction or opposite direction and may have different magnitudes.

5. Equal Vectors

Two vectors are called equal vectors if they have the same magnitude and the same direction. Their positions in space may be different, but equality depends only on magnitude and direction.

6. Negative of a Vector

A vector having the same magnitude as a given vector but opposite direction is called the negative of that vector. For example, \[\overrightarrow{BA} = -\overrightarrow{AB}\].

7. Free Vectors

Vectors that can be moved parallel to themselves without changing magnitude and direction are called free vectors.