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 June 4th, 2012, 08:14 PM #1 Member   Joined: Jun 2010 Posts: 35 Thanks: 0 The projection of the vector For the give vector $a=(3,1)$ and $b=(1,1)$, find the projection of vector $a$ in $b$. Please include a step by step solution.
June 5th, 2012, 05:49 AM   #2
Math Team

Joined: Dec 2006
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Re: The projection of the vector

Hello, Phatossi!

Quote:
 $\text{Given: }\:\vec a \:=\:\langle 3,\,1\rangle,\;\vec b \:=\:\langle 1,\,1\rangle \text{Find the projection of }\vec a\text{ onto }\vec b.$

$\text{There is a }formula \text{ for this problem.}$

[color=beige]. . [/color]$\text{proj\,}_{\vec b}\vec a \;=\;\frac{\vec a\,\cdot\,\vec b}{(|\vec b|)^2}\,\vec b$

 June 7th, 2012, 07:47 AM #3 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: The projection of the vector Draw a picture: The vectors a and b are two line segments joined at their "tails". Drop a perpendicular from the tip of a to b and you have a right triangle with hypotenuse of length |a|. Call the angle between the two vectors "$\theta$". Then the leg of that triangle, lying along vector b, has length $|a|cos(\theta)$. Since the dot product of two vectors, a and b, can be written as $a\cdot b= |a||b|cos(\theta)$, the scalar projection of a on b can be written $|a|cos(\theta)= cos(\theta)\frac{|b|}{|b|}= \frac{a\cdot b}{|b|}$. To get the vector projection of a on b, multiply that by the unit vector in the direction of b: $\frac{a\cdot b}{|b|}\frac{b}{|b|}= \frac{a\cdot b}{|b|^2} b$, the formula soroban gives.

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