We did not discuss the last property, the one involving u+v. When we talk about the resultant vector u+v being in line with the vector w, we basically are talking about how much in line the component vectors u and v are with the vector w.
I think we could also make good sense of this using the definition of inner product for functions, combining the integrals, but I am not sure if that is a good approach to justify the rules.
muxingz
For the last one, I think we could imagine that if we move the origin of vector v to the end of vector u, so to connect these two vectors and then make projection of these two vectors towards vector w. Since the connection point of these two vectors would make no contribution to the projection profile, it is just the start point of vector u and end point of v should decide the projection along vector w. And the vector (u+v) is just the one connects starting of u and ending of v.
ninkamat
Since we know that two vectors can be added by adding the principal components we can assume that w is one of the principal axes for u and v. Thus, the inner product will be the magnitude/norm of w times the sum of components of u and v along w, which is same as the magnitude/norm of w times the component of (u+v) along w.
Eg. if u = (1,2) and v = (5,2) and w is a unit vector along x axis (1,0) then,
u+v = (6,4)
(u+v).w = 6
u.w + v.w = 1 + 5 = 6
A more formal proof would be as follows,
Assuming u = (u1, u2) ; v = (v1, v2) and w = (w1, w2)
u.w = u1w1 + u2w2
v.w = v1w1 + v2w2
u + v = (u1+v1, u2+v2)
(u + v).w = w1(u1+v1) + w2(u2+v2)
(u + v).w = u1w1 + v1w1 + u2w2 + v2w2
We can re-arrange to get,
(u + v).w = (u1w1 + u2w2) + (v1w1 + v2w2)
(u + v).w = u.w + v.w
pranitd
The last property can also be explained in terms of linearity. The inner product can be assumed as a linear function in the first argument and exhibiting the additive property.
We did not discuss the last property, the one involving u+v. When we talk about the resultant vector u+v being in line with the vector w, we basically are talking about how much in line the component vectors u and v are with the vector w.
I think we could also make good sense of this using the definition of inner product for functions, combining the integrals, but I am not sure if that is a good approach to justify the rules.
For the last one, I think we could imagine that if we move the origin of vector v to the end of vector u, so to connect these two vectors and then make projection of these two vectors towards vector w. Since the connection point of these two vectors would make no contribution to the projection profile, it is just the start point of vector u and end point of v should decide the projection along vector w. And the vector (u+v) is just the one connects starting of u and ending of v.
Since we know that two vectors can be added by adding the principal components we can assume that w is one of the principal axes for u and v. Thus, the inner product will be the magnitude/norm of w times the sum of components of u and v along w, which is same as the magnitude/norm of w times the component of (u+v) along w.
Eg. if u = (1,2) and v = (5,2) and w is a unit vector along x axis (1,0) then,
u+v = (6,4)
(u+v).w = 6
u.w + v.w = 1 + 5 = 6
A more formal proof would be as follows, Assuming u = (u1, u2) ; v = (v1, v2) and w = (w1, w2)
u.w = u1w1 + u2w2
v.w = v1w1 + v2w2
u + v = (u1+v1, u2+v2)
(u + v).w = w1(u1+v1) + w2(u2+v2)
(u + v).w = u1w1 + v1w1 + u2w2 + v2w2
We can re-arrange to get,
(u + v).w = (u1w1 + u2w2) + (v1w1 + v2w2)
(u + v).w = u.w + v.w
The last property can also be explained in terms of linearity. The inner product can be assumed as a linear function in the first argument and exhibiting the additive property.