JEE Main & Advanced Mathematics Vector Algebra Types of Vector

(1) Zero or null vector : A vector whose magnitude is zero is called zero or null vector and it is represented by \[\overrightarrow{O}\].

(2) Unit vector : A vector whose modulus is unity, is called a unit vector. The unit vector in the direction of a vector \[\mathbf{a}\] is denoted by\[\mathbf{\hat{a}}\], read as \[a\text{ }cap\]. Thus, \[|\mathbf{\hat{a}}|\ =1\].

\[\mathbf{\hat{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}=\frac{\text{Vector }a\text{ }}{\text{Magnitude of }a}\]              

(3) Like and unlike vectors : Vectors are said to be like when they have the same sense of direction and unlike when they have opposite directions.

(4) Collinear or parallel vectors : Vectors having the same or parallel supports are called collinear or parallel vectors.

(5) Co-initial vectors : Vectors having the same initial point are called co-initial vectors.

(6) Coplanar vectors : A system of vectors is said to be coplanar, if their supports are parallel to the same plane.

Two vectors having the same initial point are always coplanar but such three or more vectors may or may not be coplanar.

(7) Coterminous vectors : Vectors having the same terminal point are called coterminous vectors.

(8) Negative of a vector : The vector which has the same magnitude as the vector \[\mathbf{a}\] but opposite direction, is called the negative of \[\mathbf{a}\] and is denoted by \[-\mathbf{a}\]. Thus, if \[\overrightarrow{PQ}=\mathbf{a}\], then \[\overrightarrow{QP}=-\mathbf{a}\].

(9) Reciprocal of a vector : A vector having the same direction as that of a given vector \[\mathbf{a}\] but magnitude equal to the reciprocal of the given vector is known as the reciprocal of \[\mathbf{a}\] and is denoted by \[{{\mathbf{a}}^{-1}}\]. Thus, if \[|\mathbf{a}|\,=\mathbf{a},|{{\mathbf{a}}^{-1}}|\,=\frac{1}{\mathbf{a}}\].

(10) Localized and free vectors : A vector which is drawn parallel to a given vector through a specified point in space is called a localized vector. For example, a force acting on a rigid body is a localized vector as its effect depends on the line of action of the force. If the value of a vector depends only on its length and direction and is independent of its position in the space, it is called a free vector.

(11) Position vectors : The vector \[\overrightarrow{OA}\]which represents the position of the point A with respect to a fixed point O (called origin) is called position vector of the point A. If \[(x,y,z)\] are co-ordinates of the point A, then \[\overrightarrow{OA}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\].

(12) Equality of vectors : Two vectors \[\mathbf{a}\] and \[\mathbf{b}\] are said to be equal, if (i) \[|\mathbf{a}|=|\mathbf{b}|\] (ii) They have the same or parallel support and (iii) The same sense.