Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 02/Hint 2

Let ${\hat {c}}={\frac {c}{|c|}}$. Using the picture below as a guide for a sketch of what a projection is

we can draw the following conclusions. In the image and below we assume that the angle is acute. If the angle is obtuse the final result is the same, and the argument is very similar.

1. $\cos \theta ={\frac {|{\text{Proj}}_{c}a|}{|a|}}$
2. $\cos \theta ={\frac {a\cdot c}{|a||c|}}$ (by the definition of the dot product)
3. Thus ${\frac {|{\text{Proj}}_{c}a|}{|a|}}={\frac {a\cdot c}{|a||c|}}$ and solving gives

$|{\text{Proj}}_{c}a|={\frac {a\cdot c}{|c|}}$

So now we know the length of the desired projection. The direction is c. To make sure that the length of the vector in the final answer is the one calculated above, we choose the unit vector ${\hat {c}}$ instead:

${\text{Proj}}_{c}a=\left({\frac {a\cdot c}{|c|}}\right){\hat {c}}={\frac {a\cdot c}{|c|^{2}}}c$