# Linear combination of unit vector

Linear combination of unit vector is of the form $v_{1}$ i + $v_{2}$ j . $v_{1}$ and $v_{2}$ are scalars and they are called horizontal vertical components of 'v' respectively.
Let us consider standard unit vectors $\left \langle 1,0 \right \rangle$ and $\left \langle 0,1 \right \rangle$.
i = $\left \langle 1,0 \right \rangle$ and j = $\left \langle 0,1 \right \rangle$ .This i is always written in boldface to distinguish from the imaginary number $\sqrt{-1}$.
v =$\left \langle v_{1},v_{2} \right \rangle$
= $v_{1}\left \langle 1,0\right \rangle$ + $v_{2}\left \langle 0,1\right \rangle$
=$v_{1}$
i + $v_{2}$ j
This form is called the linear combination of the vectors
i and j .
Any vector in the plane can be written as linear combination using standard unit vectors
i and j .

## Examples on linear combination of unit vector

Example 1 : Let u be the vector with initial point (3, -5) and terminal point (-1, 3). Write u as a linear combination of the standard unit vectors of i and j.
Solution : u =$\left \langle -1-3,3-(-5) \right \rangle$
=$\left \langle -4,8 \right \rangle$
∴ u = -4
i + 8 j
Graphically it is represented as follows.
Example 2 : Let u be the vector with initial point (-2, 6) and terminal point (-8, 3). Write u as a linear combination of the standard unit vectors of i and j.
Solution : u =$\left \langle -8-(-2),3-6) \right \rangle$
=$\left \langle -6,-3 \right \rangle$
∴ u = -6
i + (-3) j