# 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**

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