Question

What is the definition of vector?

Answer 1

Hint: Here we will check the two types of quantities: scalar and vector quantities with examples. We will check how a vector is represented in a plane by considering two points \[P\left({{x}_{1}},{{y}_{1}},{{z}_{1}} \right)\] and \[Q\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\]. Further we will understand the process to find the direction of a vector.

Complete step by step solution:
Here we have been asked to define a vector quantity. Let us understand this term with the help of some examples and its difference with the scalar quantity.
Now, in mathematics we generally come across two types of quantities namely: scalar quantities and vector quantities. Let us check them one by one.
(1) Scalar quantities: - A scalar quantity is a quantity that has only magnitude and no direction. For example: - mass, time, temperature, speed, work, power etc.
(2) Vector quantities: - A vector quantity is a quantity that has both magnitude and direction. For example: - weight, force, displacement, velocity, acceleration etc. Now, if we say a force of 10 Newton is acting on a body then it has no meaning but if we say the same force is acting on the body in downward direction then it makes some sense as the direction of the force is specified.
A vector is represented by an arrow $\left( \to \right)$ where the head of the arrow represents the
direction of the vector. For example: - let us consider two points in a plane given as \[P\left(
{{x}_{1}},{{y}_{1}},{{z}_{1}} \right)\] and \[Q\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\] then the vector from
P to Q will be denoted as $\overrightarrow{PQ}$ and is given as:
$\Rightarrow \overrightarrow{PQ}=\left( {{x}_{2}}-{{x}_{1}} \right)\hat{i}+\left( {{y}_{2}}-{{y}_{1}}
\right)\hat{j}+\left( {{z}_{2}}-{{z}_{1}} \right)\hat{k}$
Here, $\hat{i},\hat{j},\hat{k}$ represents the unit vectors along the positive x, y, z direction respectively. They are called unit vectors because their magnitude is 1 according to the convention.

Note: Note that if you want to write a vector from Q to P then simply multiply both the sides of $\overrightarrow{PQ}$ with -1. The operation will reverse the direction of the given vector and we will get $\overrightarrow{QP}=\left( {{x}_{1}}-{{x}_{2}} \right)\hat{i}+\left( {{y}_{1}}-{{y}_{2}}\right)\hat{j}+\left( {{z}_{1}}-{{z}_{2}} \right)\hat{k}$. Vectors are a very important topic used in Physics so you must have the basic idea regarding the basic terms.