what actually happens when two vectors are multiplied?
I understand that in scalar form it is a quantity times a ratio...
but how the heck do angles multiply together along with magnitude??
i am a visual learner, this kind of really conceptual thing is very difficult for me...
someone please explain what the vector dot product actually IS? (i know its got something to do with proving two vectors are perpendicular)
thnx
well, it's hard to really explain most of this. usually the best way is to grab a spec book and read about it.
but anyway. my personal understanding of the dot product is that it's a way of telling you how large each vector is, and how large the angle is between them.
that is, for two vectors
and
, which are an angle
apart, we have
but you already knew that, of course.
perhaps you might think of the dot product as a function.
suppose
and
are
n-dimensional vectors: that is,
then the dot product
can be thought of as
: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R})
so the dot product is a function that has an input of two vectors, and gives an output of a scalar.
the main point of the dot product is that it is commutative, distributive, and that it has the concept of orthogonality (that is, it shows whether two vectors are perpendicular to each other).
that is, the dot product has the following important properties:
(commutativity)
 = u \cdot v + u \cdot w)
(distributivity)
c \cdot v = )
c)
, where
c is a scalar.
if and only if
for
non-zero,
if and only if
\pi / 2)
, with
- that is, if the two vectors are perpendicular.
the last property is the one that is so important, and is really the main point of ever using the dot product, i guess.
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