# Vector Operation

In this vector operation section, ask-math will explain about the scalar multiplication.
Geometrically, the product of vector
v and a scalar k is the vector that |k| times vector v . The direction of the new vector 'kv' depends on the value of 'k'. If k is positive then the direction of $\vec{kv}$ to that of $\vec{v}$ and if k is negative then $\vec{kv}$ will be in the opposite direct to that of $\vec{v}$.
According to above diagram, in 2nd diagram the value of k is positive so the direction of $\vec{1/2v}$ is same as the direction of $\vec{v}$. In the 3rd diagram the magnitude of $\vec{-1/2v}$ is same as $\vec{1/2v}$ but here k is negative so the direction of $\vec{-1/2v}$ is opposite to the direction of $\vec{1/2v}$

## Examples on Vector Operation

Example :1 v =$\left \langle-3,5 \right \rangle$. Find a) 2v b) $\frac{3}{2}$
Solution : a) As v = $\left \langle-3,5 \right \rangle$

So 2v = 2$\left \langle-3,5 \right \rangle$

= $\left \langle2(-3),2(5) \right \rangle$ = $\left \langle-6,10 \right \rangle$

b) As v = $\left \langle-3,5 \right \rangle$

So $\frac{3}{2}$v =$\frac{3}{2}\left \langle-3,5 \right \rangle$

= $\left \langle\frac{3}{2}\times -3,\frac{3}{2}\times 5 \right \rangle$

= $\left \langle\frac{-9}{2},\frac{15}{2} \right \rangle$

Example :2 Let u =$\left \langle 1,4 \right \rangle$ v =$\left \langle 3,2 \right \rangle$. Find 2u - 3v
Solution : As u = $\left \langle 1,4 \right \rangle$ and v = $\left \langle3,2 \right \rangle$

So 2u = 2$\left \langle 1,4 \right \rangle$ = $\left \langle 2,8 \right \rangle$

and 3v = 3$\left \langle 3,2 \right \rangle$ =$\left \langle 9,6 \right \rangle$

So 2u - 3v = $\left \langle 2 - 9,8 - 6 \right \rangle$

2u - 3v =$\left \langle -7,2 \right \rangle$