Vectors: Dot Product

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Vectors: Dot Product

April 19th, 2010 admin Leave a comment Go to comments

The dot product or scalar product is a way on combining two vectors to get a real number.

The dot product is defined to be

$a \cdot b = |a||b|cos(\theta)$ where $\theta$ is the angle between the vectors a and b.

things to notice:

if a and b are parallel $cos(\theta)=1$ so the dot product is simply the product of their lengths
if and and b are perpendicular $cos(\theta)=0$ so the dot product is 0
$a \cdot b = b \cdot a$
if a is a unit vector, is |a|= 1 then the dot product gives the magnitude of the component of b in the direction of a

Dot product in component form (with i,j,k’s)

The dot product is very easy to use in component form because of 1 and 2. Since i,j and k are all perpendicular to each other:
$i \cdot j = i \cdot k = j \cdot k = 0$
and since any vector is parallel to itself and i,j and k are all unit vectors
$i \cdot i = j \cdot\j = k \cdot k =1$

hence we can find the dot product of $a = a_{1}i + a_{2}j+a_{3}k$ with $b=b_{1}i+b_{2}j+b_{3}k$
Now we simply take the dot product of each term in a with each term of b, in a similar way to how you multiply out brackets, to get

$a \cdot b = a_{1}b_{1}i\cdot i + a_{1}b_{2}i\cdot j + a_{1}b_{3}i\cdot k+ a_{2}b_{1}j\cdot i+ a_{2}b_{2}j\cdot j+ a_{2}b_{3}j\cdot k+ a_{3}b_{1}k\cdot i + a_{3}b_{2}k\cdot j + a_{3}b_{3}k\cdot k$ and substituting in the above equations for the dot products of i,j,k we get
$a \cdot b = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}+$

Trevor Pythagoras Maths