The dot product of two vectors is the sum of the product of each of the components of two vectors.
First, observe that for any vector A,
which is the squared length of A
Also, the dot product is commutative ...
... distributive (with respect to vector addition) ...
... and associative (with respect to scalar multiplication) ... or perhaps bilinear as suggested by the first comment down below ...
Next time I'll show some more interesting properties of the dot product.
Glad to see you posting again. I had almost given up...and I like where I think you're going with this sequence of posts. A few minor comments:
ReplyDeleteA\dot(B+C)=A\dotB+A\dotC is the distributive law.
Calling t(A\dotB)=(tA)\dotB=A\dot(tB) the associative law is a little wierd, too, because there are two operations (scalar mult. and dot product). Associativity generally refers to a single operation and goes (AB)C=A(BC). The dot product is not associative, but scalar multiplication is.
If you really want a word, you can combine these two rules into one called "bilinearity."
(A+B)\dot C=A\dot C + B\dot C
A\dot(B+C)= A\dot B+ A\dot C
t(A\dot B)= (tA)\dot B = A\dot(tB)
Then the dot product is a commutative bilinear operation.
At last!
ReplyDeleteWe actually skipped this part in me math class, and I couldn't reverse engineer where he was getting this "dot product" from!
Mucho gracias!
what would it mean if it was between like to values with powers. in short is it multiplication addition or what?
ReplyDelete8^x(dot)2^4x
equals
2^7x how is this possible