Prove that u=5i-4j and v=2i+3j are closing an obtuse angle.

If the angle between the vectors u and v is obtuse, then
the value of cosine of the angle between u and v must be within the interval
(-1,0).

We'll calculate the cosine of the angle from the
dot product between and .

We'll recall the
formula that gives the dot product of   and
:

* = | *| |*cos(
)

We'll calculate the product of
vectors:

(5i-4j)(2i+3j)

4*2*vecj*veci - 4*3*vecj^2

|veci|*|veci|*cos 0

But
1

0

-2

4^2)

sqrt41

3^2)

We'll
calculate the cosine between the vectors u and v:

cos(
, vecv(vecu*vecv)/(|vecu||vecv|)

-2/sqrt533 < 0

Since the value of
cosine angle is between the interval (-1,0) the angle closed by the vectors u and v is
obtuse.