Notes On Types of Vectors - CBSE Class 12 Maths
A vector whose initial and terminal points are the same is called a zero vector. Zero Vector $\stackrel{\to }{\text{AA}}$ = $\stackrel{\to }{\text{0}}$ ⎢ $\stackrel{\to }{\text{0}}$ ⎢ = 0 A zero vector has no fixed direction. A vector whose magnitude is 1 is called a unit vector. ⎢$\stackrel{\to }{\text{AB}}$ ⎢ = 1 A unit vector is represented as ⎢$\stackrel{^}{\text{a}}$⎢ = 1 Two vectors having the same initial point are called coinitial vectors. l  || ⎢$\stackrel{\to }{\text{AB}}$ ⎢ ⎢ ⎢ || ⎢ ⎢ || ⎢ ⎢ || l Two or more vectors that are parallel to the same line are called collinear vectors. Collinear vectors may have different magnitudes. Collinear vectors may have opposite directions. $\stackrel{\to }{\text{AB}}$ and have the same direction. ⎢$\stackrel{\to }{\text{AB}}$ ⎢ = ⎢$\stackrel{\to }{\text{OP}}$ ⎢ Two vectors with the same direction and magnitude are called equal vectors. $\stackrel{\to }{\text{AB}}$ = $\stackrel{\to }{\text{OP}}$ Equal vectors may have different initial and terminal points. ⎢$\stackrel{\to }{\text{AB}}$ ⎢ = ⎢$\stackrel{\to }{\text{OP}}$ ⎢ $\stackrel{\to }{\text{AB}}$ and $\stackrel{\to }{\text{OP}}$ are in opposite directions. Two vectors with the same magnitude but opposite directions are called negative vectors of each other. Ex: $\stackrel{\to }{\text{PO}}$ = – $\stackrel{\to }{\text{AB}}$ = $\stackrel{\to }{\text{BA}}$ $\stackrel{\to }{\text{AB}}$  = – $\stackrel{\to }{\text{PO}}$ = $\stackrel{\to }{\text{OP}}$ Note : All equal vectors are collinear vectors,But all collinear vectors are not equal. A pair of collinear vectors may be negative vectors of each other. Suppose change the position of vector AB such that its direction and magnitude do not change.Then, in each new position, vector AB is parallel to its initial position and represents the same vector. A vector that is independent of its position is called a free vector. Thus, a free vector is not bound by any fixed initial or terminal points.

#### Summary

A vector whose initial and terminal points are the same is called a zero vector. Zero Vector $\stackrel{\to }{\text{AA}}$ = $\stackrel{\to }{\text{0}}$ ⎢ $\stackrel{\to }{\text{0}}$ ⎢ = 0 A zero vector has no fixed direction. A vector whose magnitude is 1 is called a unit vector. ⎢$\stackrel{\to }{\text{AB}}$ ⎢ = 1 A unit vector is represented as ⎢$\stackrel{^}{\text{a}}$⎢ = 1 Two vectors having the same initial point are called coinitial vectors. l  || ⎢$\stackrel{\to }{\text{AB}}$ ⎢ ⎢ ⎢ || ⎢ ⎢ || ⎢ ⎢ || l Two or more vectors that are parallel to the same line are called collinear vectors. Collinear vectors may have different magnitudes. Collinear vectors may have opposite directions. $\stackrel{\to }{\text{AB}}$ and have the same direction. ⎢$\stackrel{\to }{\text{AB}}$ ⎢ = ⎢$\stackrel{\to }{\text{OP}}$ ⎢ Two vectors with the same direction and magnitude are called equal vectors. $\stackrel{\to }{\text{AB}}$ = $\stackrel{\to }{\text{OP}}$ Equal vectors may have different initial and terminal points. ⎢$\stackrel{\to }{\text{AB}}$ ⎢ = ⎢$\stackrel{\to }{\text{OP}}$ ⎢ $\stackrel{\to }{\text{AB}}$ and $\stackrel{\to }{\text{OP}}$ are in opposite directions. Two vectors with the same magnitude but opposite directions are called negative vectors of each other. Ex: $\stackrel{\to }{\text{PO}}$ = – $\stackrel{\to }{\text{AB}}$ = $\stackrel{\to }{\text{BA}}$ $\stackrel{\to }{\text{AB}}$  = – $\stackrel{\to }{\text{PO}}$ = $\stackrel{\to }{\text{OP}}$ Note : All equal vectors are collinear vectors,But all collinear vectors are not equal. A pair of collinear vectors may be negative vectors of each other. Suppose change the position of vector AB such that its direction and magnitude do not change.Then, in each new position, vector AB is parallel to its initial position and represents the same vector. A vector that is independent of its position is called a free vector. Thus, a free vector is not bound by any fixed initial or terminal points.

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