Vectors
1.0 Introduction
2.0 Types of Vectors
3.0 Addition of Vectors
4.0 Components of a Vector
5.0 Vector Joining Two Points
6.0 Projection of a Vector on a Line
7.0 Section Formula
8.0 Products of a Vector
9.0 Lami's Theorem
10.0 Linear Combination of Vectors
11.0 Linearly Dependent and Independent Vectors
12.0 Scalar Triple Product
13.0 Vector Triple Product
3.2 Properties of Vector Addition
Property 1: $\overrightarrow p + \overrightarrow q = \overrightarrow q + \overrightarrow p $ (commutative property)Proof:
Consider the parallelogram $PQRS$ as shown in figure. Let $\overrightarrow {PQ} = \overrightarrow p $ and $\overrightarrow {QR} = \overrightarrow q $ then using the triangle law, from triangle $PQR$, we have $\overrightarrow {PR} = \overrightarrow p + \overrightarrow q $ but, we know that the opposite sides of a parallelogram are equal and parallel, from figure, we have, $$\overrightarrow {PS} = \overrightarrow {QR} = \overrightarrow q $$ and $$\overrightarrow {SR} = \overrightarrow {PQ} = \overrightarrow p $$ Again using triangle law, from triangle $PSR$, we have $$\overrightarrow {PR} = \overrightarrow {PS} + \overrightarrow {SR} = \,\overrightarrow q + \overrightarrow p $$$$\overrightarrow p + \overrightarrow q = \overrightarrow q + \overrightarrow p $$
Property 2: $\overrightarrow {(p} + \overrightarrow q ) + \overrightarrow r = \overrightarrow p + (\overrightarrow q + \overrightarrow r )$ (associative property)Proof:
Let $PQ$, $QR$ and $RS$ be represented by the vectors $\overrightarrow p $, $\overrightarrow q $ and $\overrightarrow r $ respectively, as shown in figure.
$$\begin{equation} \begin{aligned} \overrightarrow p + \overrightarrow q = \overrightarrow {PQ} + \overrightarrow {QR} = \overrightarrow {PR} \\ \overrightarrow q + \overrightarrow r = \overrightarrow {QR} + \overrightarrow {RS} = \overrightarrow {QS} \\ (\overrightarrow p + \overrightarrow q ) + \overrightarrow r = \overrightarrow {PR} + \overrightarrow {RS} = \overrightarrow {PS}...(i) \\ \overrightarrow p + \overrightarrow {(q} + \overrightarrow r ) = \overrightarrow {PQ} + \overrightarrow {QS} = \overrightarrow {PS}...(ii) \\\end{aligned} \end{equation} $$
Property 3: $\overrightarrow p + \overrightarrow 0 = \overrightarrow p $ (additve property)
Proof:
Using commutative property,$$\overrightarrow p + \overrightarrow 0 = \overrightarrow 0 + \overrightarrow p $$
$0$ vector is called additive identity
Property 4: $\overrightarrow p + ( - \overrightarrow p ) = \overrightarrow 0 $ (additive inverse)
Proof:
When $k = –1$, then $k\overrightarrow p = - \overrightarrow p $, which is a vector having magnitude equal to the magnitude of vector $p$ and direction is opposite to that of the direction of vector $p$. The vector $ - \overrightarrow p $ is called the negative (or additive inverse) of vector $\overrightarrow p $ and we always have
$$\vec p + ( - \vec p) = \vec 0\quad ({\text{additive inverse)}}$$$$=( - \overrightarrow p ) + (\overrightarrow p ) = \overrightarrow 0 $$
Property 5: $({k_1} + {k_2})\overrightarrow p = {k_1}\overrightarrow p + {k_2}\overrightarrow p $ (multiplication by scalars)
Proof:
When a vector $\overrightarrow p $ is multiplied by any scalar $(k)$, resultant is also a vector which is having same direction but magnitude is different and hence the new vector is parallel to initial vector. Magnitude of new vector is $k$ times of vector, where $k$ is a scalar (which can be fraction, integer etc).
Property 6: $k(\overrightarrow p + \overrightarrow q ) = k\overrightarrow p + k\overrightarrow q $
Property 7: $k(m\overrightarrow {p)} = (km)\overrightarrow p $ where $k$ and $m$ are scalars.
