find a unit vector and a vector of magnitude 3 in the opposide direction

**alessandromangione** · January 27th 2010, 07:37 PM

seeing:
A(4,0) B(2,1)

find a unit vector in the direction of AB ( smal arrow above AB that begins from left to right) abd a vector of magnitude 3 in the opposite direction

thank you

**fishcake** · January 27th 2010, 08:09 PM

A unit vector is simply a vector with a magnitude of 1. To get the unit vector in the direction of the vector $\overrightarrow{A B}$ , all you have to do is divide the components of the vector by its magnitude. This process is called normalization.

$\vec{AB}$
$= (2 - 4, 1 - 0)$
$= (-2, 1)$

Magnitude of $\vec{AB}$
$= \sqrt {{x}^{2}+{y}^{2}}$
$= \sqrt {{(-2)}^{2}+{1}^{2}}$
$= \sqrt {5}$

Unit vector in the direction of $\vec{AB}$
$= \left ( \frac{-2}{\sqrt{5}}, \frac{1}{\sqrt{5}} \right )$

To find the vector of magnitude 3 in the opposite direction of the vector $\vec{AB}$ , you have to find the unit vector in the direction of $\vec{BA}$ , which is in the opposite direction. Since the unit vector has a magnitude of 1, and you need a vector of magnitude 3, just multiply the unit vector by 3, and you'll get the answer.

**alessandromangione** · January 27th 2010, 08:23 PM

thank u very much!

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