# How do you find the component form and magnitude of the vector v given initial point (1,3) and terminal point (-8,-9)?

Mar 28, 2017

Component form is v $= < - 9 , - 12 >$
Magnitude is ||v|| = $15$

#### Explanation:

To find the component form, you only need to know how to substitute figures for letters. What do I mean by this?

If initial side is $\left({x}_{1} , {y}_{1}\right)$

Then ${x}_{1} = 1$ and ${y}_{1} = 3$

If terminal side is $\left({x}_{2} , {y}_{2}\right)$

Then ${x}_{2} = - 8$ and ${y}_{2} = - 9$

Thus, component form of v is $< \left({x}_{2} - {x}_{1}\right) , \left({y}_{2} - {y}_{1}\right) >$...simply $< x , y >$

In this case, v $= < \left[\left(- 8\right) - 1\right] , \left[\left(- 9\right) - 3\right]$

Which gives us v $= < - 9 , - 12 >$

To find the magnitude of a vector, the concept of pythagorean theorem needs to be understood.

Why?

If v$= < x , y >$, using pythagorean theorem, ||v||$= \sqrt{{x}^{2} + {y}^{2}}$

We already know the values for v, so;

||v|| = $\sqrt{{\left(- 9\right)}^{2} + {\left(- 12\right)}^{2}}$

||v|| = $\sqrt{81 + 144}$

||v|| = $\sqrt{225}$

||v||$= 15$