A) show geometrically that, for any scalar k and any vectors u and v,
B) illustrate for k>0 that k(u+v)=ku+kv
Note: bolded are vectors (arrows on top)
ok so these two questions are quite similar.. I think, but I am still having trouble on "proving" them. I am not sure if my method is O.K. (no answers in back of book), so can someone please verify or demonstrate the correct method. Thanks in advance
is this ok? does it prove the q?
B) well I thought about this as for any constant number k, it has to be greater than 0 because otherwise the resultant will be 0 and does not prove ku+kv
what about this question?
ok, ill try to describe my drawings
u and v are vectors |u|=|v|=1
not sure how to draw the k part but I drew the (u-v) first
ill describe as a triangle: base is u, v is hypotenuse, (u-v) is height
B) what does it mean by k>0 that k(u+v)=ku+kv
how are u supposed to draw that? doesnt it mean that
ku+kv=ku+kv? im not sure how i would draw these to prove this eqtn
To prove A)
Draw two vectors, u and v.
Draw the vector u - v.
Draw a vector parallel to u - v, of a different length (k).
This vector is k(u - v).
Then draw the vectors ku and kv.
Draw the vector ku - kv.
It should be clear that the vectors k(u - v) and ku - kv are parallel and of the same length. Therefore they are equal.
The same process is used to prove B).
ok, i think I have part A) thanks,
for part B), what does it mean by k>0..
im not sure how to SHOW k>0 by drawing
like.. does it mean that k must exist, for k>0, or, does it mean that -k (cant go in a diff direction) is not allowed?
so I have:
draw vectors ku and kv
then draw ku+kv vector
draw vectors u and v, then draw vector (u+v), then beside it (parallel) draw k(u+v) but of diff length.
so now my question is, how do you know about the k>0 does it mean k must exist or does it mean that k cannot be going in a diff direction