We can only consider the inner product to be the projection from one to another when the two are both unit vector, right?
UhrmasJHHue
But wait, if the length of v is unchanged and the length of u is doubled, wouldn't the projection of v onto u remain unchanged? Like isn't it still |v|*cosx (x being the angle between u and v)? And only the projection of u onto v would be doubled?
Or, is it like, if we extend the length of one vector, when we calculate the inner product we would also extend the other vector by the same ratio and then look at the projection?
CMUScottie
@UhrmasJHHue I think the inner product measures the length of projected vector only when the vector is unit vector.
tarangs
I guess the second part of the slide is confusing.
The first half of the slide mentions that the inner product is a "projection" only when the vectors are unit vectors.
When we scale the vectors, the condition that the 2 vectors are "unit" vectors fails. So though in isolation, the fact that "the inner product scales as one of the vectors is scaled" is true. The projection analogy on the top half is not meant for this condition.
In the lecture video, professor says that "If <u,v> is the size of the shadow of V on u, the doubling the length of V will double the length of the shadow on u as well"
atarng
This works iff the vector being projected on is a unit vector, right? It should work even if the projected vector is not a unit vector, due to the second half of the slide
tib
Agreeing with @tarangs, I think the two parts of the slide should be treated separately, given that the second part does not satisfy the requirement where both vectors are unit vectors.
We can only consider the inner product to be the projection from one to another when the two are both unit vector, right?
But wait, if the length of v is unchanged and the length of u is doubled, wouldn't the projection of v onto u remain unchanged? Like isn't it still |v|*cosx (x being the angle between u and v)? And only the projection of u onto v would be doubled?
Or, is it like, if we extend the length of one vector, when we calculate the inner product we would also extend the other vector by the same ratio and then look at the projection?
@UhrmasJHHue I think the inner product measures the length of projected vector only when the vector is unit vector.
I guess the second part of the slide is confusing.
The first half of the slide mentions that the inner product is a "projection" only when the vectors are unit vectors.
When we scale the vectors, the condition that the 2 vectors are "unit" vectors fails. So though in isolation, the fact that "the inner product scales as one of the vectors is scaled" is true. The projection analogy on the top half is not meant for this condition.
In the lecture video, professor says that "If <u,v> is the size of the shadow of V on u, the doubling the length of V will double the length of the shadow on u as well"
This works iff the vector being projected on is a unit vector, right? It should work even if the projected vector is not a unit vector, due to the second half of the slide
Agreeing with @tarangs, I think the two parts of the slide should be treated separately, given that the second part does not satisfy the requirement where both vectors are unit vectors.