# How do you find the component form and magnitude of the vector v given initial point (-1,5) and terminal point (15, 12)?

Mar 24, 2017

Component form is v $= < 16 , 7 >$
Magnitude is ||v|| = $\sqrt{305} \approx 17.46$

#### 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 ${y}_{1} = 5$

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

Then ${x}_{2} = 15$ and ${y}_{2} = 12$

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 $= < \left[15 - \left(- 1\right)\right] , \left(12 - 5\right)$

Which gives us v $= < 16 , 7 >$

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(16^2+7^2

||v|| = $\sqrt{256 + 49}$

||v|| = $\sqrt{305}$

||v||$\approx 17.46$