# Adding and Subtracting Algebraic Vectors

Adding vectors results in a new Resultant Vector. Geometric and algebraic vectors can be added using different techniques.

Algebraic vectors can be added by adding the respective components along each axis:

$$\vec{a} = \left<3,4\right>, \; \vec{b} = \left<-4,10\right>$$

$$\vec{a} + \vec{b} = \left<3 + -4, 4 + 10\right> = \left<-1, 14\right>$$

 Commutative Law of Addition $$\vec{u} + \vec{v} = \vec{v} + \vec{u}$$ Associative Law of Addition $$(\vec{m} + \vec{n}) + \vec{o} = \vec{m} + (\vec{n} + \vec{o})$$

If $$\vec{d} = \left<3,6\right>$$ and $$\vec{e} = \left<-4,-5\right>$$ find $$2 \vec{d} + 3 \vec{e}$$.

## Subtracting Algebraic Vectors

Like adding Algebraic vectors, subtract the respective components along each axis:

$$\vec{a} = \left<3,4\right>, \; \vec{b} = \left<-4,10\right>$$

$$\vec{a} - \vec{b} = \left<3 - (-4), 4 - 10\right> = \left<7, -6\right>$$

You can think of subtracting vectors as adding the Opposite vector:

$$\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$$

$$= \left<3,4\right> + (-\left<-4,10\right>)$$

$$= \left<3,4\right> + \left<4,-10\right>$$

$$= \left<3+4,4-10\right>$$

$$= \left<7,-6\right>$$