I think we discussed all but the last property in class.

The last property makes intuitive sense if you think about projecting the sum of two vectors (u,v) onto a third (w) in 2D: You can do the projection normally, or you can recenter v onto w, which doesn't change v's projected length because you only moved v along a line perpendicular to w. We get the same total projected length onto w either way, so we recover the above identity.

keenan

@zbp Right. And we may not have talked about the second to last one: intuitively, if a vector gets a times longer, then its "shadow" also gets a times longer.

I think we discussed all but the last property in class.

The last property makes intuitive sense if you think about projecting the sum of two vectors (u,v) onto a third (w) in 2D: You can do the projection normally, or you can recenter v onto w, which doesn't change v's projected length because you only moved v along a line perpendicular to w. We get the same total projected length onto w either way, so we recover the above identity.

@zbp Right. And we may not have talked about the second to last one: intuitively, if a vector gets

atimes longer, then its "shadow" also getsatimes longer.