Vectors
2.0 Types of Vectors
2.0 Types of Vectors
1. Zero or Null Vector: A vector whose initial and terminal points are coincident is called zero or null vector. It is denoted by $\overrightarrow 0 $. Zero vector does not have any definite direction. It can have any direction or direction of zero vector is arbitrary. The vectors $\overrightarrow {AA} $, $\overrightarrow {BB} $ represent the zero vector.
2. Unit Vector: A vector whose magnitude is unity is called unit vector which is denoted by $\widehat {a}$ and read as a cap.
3. Coinitial Vectors: Two or more vectors having the same initial point are called coinitial vectors.
4. Collinear or parallel Vectors: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of theirmagnitudes and directions or the vectors having the same or parallel support are called collinear vectors.5. Free Vectors: If the initial point of a vector is not specified, then it is said to be a free vector.
Note: In mathematics, we mainly deal with free vectors.
6. Negative of a Vector: A vector having the same magnitude as that of a given vector $a$ and the direction opposite to that of $a$ is called the negative of $a$ and it is denoted by $ - a$. For example, vector $\overrightarrow {BA} $ is negative of the vector $\overrightarrow {AB} $ and written as $\overrightarrow {BA} $ = -$\overrightarrow {AB} $.
7. Like and Unlike Vectors: Vectors are said to be like when they have the same direction and unlike when they are in opposite direction. 8. Coterminous Vectors: Vectors having the same terminal point are called coterminous vectors.
9. Equal Vectors: Two vectors $a$ and $b$ are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as $\overrightarrow a $ = $\overrightarrow b $.
10. Localized Vectors: A vector which is drawn parallel to a given vector through a specified point in space is called localized vector.
11. Coplanar Vectors: A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Otherwise they are called non-coplanar vectors.
12. Reciprocal of a Vector: A vector having the same direction as that of a given vector but magnitude equal to the reciprocal of the given vector is known as the reciprocal of $a$ i.e., if $\left| a \right| = a$, then $\left| {{a^{ - 1}}} \right| = \frac{1}{{\left| {\overrightarrow a } \right|}}$.
Question 1. Represent graphically a displacement of $40$ km, ${45^0}$ east of north.Solution: The vector $\overrightarrow {OP} $ represents the required displacement.
Question 2. Classify the following measures into scalars and vectors.
i) $5$ seconds
ii) $2000$ $c{m^3}$
iii) $20N$
iv) $15{\text{ km/hr}}$
v) $35{\text{ m/sec}}$ towards west
vi) $30{\text{ km/hr}}$ towards east.
Solution:
| Scalar | Vector |
| time $5$ seconds | force $20N$ |
| volume $2000$$ $$c{m^3}$ | velocity $35{\text{ m/sec}}$ towards west |
| speed $15{\text{ km/hr}}$ | velocity $30{\text{ km/hr}}$ towards east |
