Vector dot product problem?
To my understanding...
a . b = |a| |b| cos(theta)
Where "a" and "b" are both vectors and theta is the angle between them.
My question:
My teacher defined the vector dot product as the vector "a" multiplied by the projection of vector "b" onto "a".
The projection of "b" is a vector, and "a" is a vector, so why is the product not a vector?
Update:"that's the definition" is not an answer.
Comments
Quote:
"My teacher defined the vector dot product as the vector "a" multiplied by the projection of vector "b" onto "a".
The projection of "b" is a vector, and "a" is a vector, so why is the product not a vector?"
I would say the definition as given by your teacher is a bit off. The dot product is the MAGNITUDE of the vector "a" multiplied by the MAGNITUDE of the projection of vector "b" onto "a".
Imagine drawing a dashed line perpendicular from "b" to the tip of "a". Now you have a right triangle formed where the projection of "b" is one of the legs. How do you find the length of a leg in a right triangle? You multiply the hypotenuse by the cos of the angle (which is equal to |b|*cos(theta)).
So... what all those words end up looking like in math.
a.b = |a|*|b|*cos(theta)
where |a| is a scalar and |b|*cos(theta), is a scalar.
So, I think you are asking a good question, because if you quoted the instructor correctly, his/her terminology was a bit off. Any more depth needed, let me know.
By DEFINITION where a and b are VECTORS and IaI and IbI are SCALARS and Ó¨ is angle between a and b :-
a • b = i a I I b I cos Ó¨
Everything on R H S is a scalar quantity thus a • b is a SCALAR PRODUCT. and is defined as such.