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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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 `vecu ` and `vecv` .

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

`vecu` *`vecv` = |`vecu|` *|`vecv` |*cos(`vecu,vecv` )

We'll calculate the product of vectors:

`vecu*vecv = (5i-4j)(2i+3j)`

`vecu*vecv = 5*2*veci^2 + 5*3*veci*vecj - 4*2*vecj*veci - 4*3*vecj^2`

`veci^2 = veci*veci = |veci|*|veci|*cos 0`

But `|veci| = |vecj| = 1`

`veci*vecj = |veci|*|vecj|*cos90 = 0`

`vecu*vecv = 10 - 12 = -2`

`|vecu| = sqrt(5^2 + 4^2)`

`|vecu| = sqrt41`

`|vecv| = sqrt(2^2 + 3^2)`

`|vecv| = sqrt13`

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

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

`cos(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.

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