# operations with vectors

**To add two vectors**, *u* and *v*, we join the extreme of *u* with the origin of *v* and then, *u + v* has the origin of *u* as its origin and the extreme of *v* as its extreme. The coordinates are the addition of their coordinates.

Example: (2,2) + (5,2) = (7,4)

**The opposite of a vector** *v*, is another vector, *-v*, with the same magnitude and direction but opposite sense. The coordinates are the opposite of v-coordinates.

Example: if v(3,-1), then -v = (-3,1)

**To subtract two vectors**, *u* and *v*, we add *u* and *–v*. The coordinates are the subtraction of their coordinates.

Example: (2,3)-(4,1) = (-2,2)

**The multiplication of a vector v by a scalar λ** (λЄR), is another vector, λ

*v*, with:

- magnitude:|λ|·|

*v*|

- the same direction with the same sense if λ>0, and opposite sense if λ<0.

The coordinates are the v-coordinates λ times.

Example: if v(3,-1), then -3v = (-9,3)

**Exercise**: If *u*(3,-2), *v*(-3,3), calculate:

a) *u + v*

b)* u - v*

c) *-4u*

* *

* *

Solutions:a) (0,1); b) (6,-5); c) (-12,8)

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