# Properties of vector

The following are some of the properties of vector addition and multiplication.
Let u and v and w be vectors and let c and d are scalars.
Property 1 : Commutative property : v + u = u + v
Example : u = $\left \langle -1,2 \right \rangle$ and v = $\left \langle 4,2 \right \rangle$
u + v = $\left \langle -1,2 \right \rangle + \left \langle 4,2 \right \rangle$ = $\left \langle 3,4 \right \rangle$
v + u = $\left \langle 4,2 \right \rangle + \left \langle -1,2 \right \rangle$ = $\left \langle 3,4 \right \rangle$
u + v = v+ u

Property 2 : Associative property (u + v)+ w = u + (v + w)
Example : u = $\left \langle -1,2 \right \rangle$ , v = $\left \langle 4,2 \right \rangle$ andw = $\left \langle 3,-1 \right \rangle$
(u + v)+ w=( $\left \langle -1,2 \right \rangle$ + $\left \langle 4,2 \right \rangle$) + $\left \langle 3,-1 \right \rangle$
= $\left \langle 3,4 \right \rangle$ + $\left \langle 3,-1 \right \rangle$
=$\left \langle 6,3 \right \rangle$
u + ( v + w)= $\left \langle -1,2 \right \rangle$ + ($\left \langle 4,2 \right \rangle$) + $\left \langle 3,-1 \right \rangle$)
= $\left \langle -1,2 \right \rangle$ + $\left \langle 7,1 \right \rangle$
=$\left \langle 6,3 \right \rangle$
∴ (u + v)+ w = u + (v + w)

Property 3: u + 0 = u
Example : u = $\left \langle -1,2 \right \rangle$
u + 0 = $\left \langle -1,2 \right \rangle + \left \langle 0,0 \right \rangle$ = $\left \langle -1,2 \right \rangle$
u + 0 = 0

Property 4: u + (-u) = 0
Example : u = $\left \langle -1,2 \right \rangle$
u + (-u) = $\left \langle -1,2 \right \rangle + -\left \langle -1,2 \right \rangle$
= $\left \langle -1,2 \right \rangle$ + $\left \langle 1,-2 \right \rangle$
=$\left \langle 0,0 \right \rangle$
u + (-u) = 0

Property 5: c(du) =(cd)(u)
Example : u = $\left \langle -1,2 \right \rangle$ , c= 2 and d= -1
c(du) = 2($\left \langle -1(-1,2) \right \rangle$
=2 $\left \langle 1,-2 \right \rangle$
= $\left \langle 2,-4\right \rangle$
(cd)u = (2 X -1)($\left \langle -1,2 \right \rangle$
=-2 $\left \langle -1,2 \right \rangle$
= $\left \langle 2,-4\right \rangle$
c(du) =(cd)(u)

## Properties of Vector

Property 6: (c + d)u = cu + du
Example : u = $\left \langle -1,2 \right \rangle$ , c= 2 and d= 3
(c + d)u = (2 + 3)($\left \langle -1,2 \right \rangle$)BR> = 5$\left \langle -1,2 \right \rangle$ = $\left \langle -5,10 \right \rangle$
cu + du = 2 $\left \langle -1,2 \right \rangle$ + 3$\left \langle -1,2 \right \rangle$
= $\left \langle -2,4 \right \rangle$ + $\left \langle -3,6 \right \rangle$
= $\left \langle -5,10 \right \rangle$
(c + d)u = cu + du

Property 7: c(u + v) = cu + cv
Example : u = $\left \langle -1,2 \right \rangle$ v = $\left \langle 4,2 \right \rangle$ and c= 2
c(u + v) = 2 ($\left \langle -1,2 \right \rangle$ + $\left \langle 4,2 \right \rangle$ )
= 2($\left \langle 3,4 \right \rangle$ )
= $\left \langle 6,8 \right \rangle$
cu + cv = 2($\left \langle -1,2 \right \rangle$) + 2($\left \langle 4,2 \right \rangle$
= $\left \langle -2,4 \right \rangle$ + $\left \langle 8,4 \right \rangle$
= $\left \langle 6,8 \right \rangle$
c(u + v) = cu + cv

Property 8: 1(u) = u
Example : u = $\left \langle -1,2 \right \rangle$
1(u) = 1($\left \langle -1,2 \right \rangle$)= $\left \langle -1,2 \right \rangle$
1(u) = u

Property 9: 0(u) = 0
Example : u = $\left \langle -1,2 \right \rangle$
0(u) = 0($\left \langle -1,2 \right \rangle$)= $\left \langle 0,0 \right \rangle$ = 0
0(u) = 0

Property 10: $\left \| cv \right \| =|c|.\left \| v \right \|$