Answer to Question #208120 in Analytic Geometry for Faith

Question

Determine whether u and v are orthogonal vectors,make an acute or obtuse angle:

1.u=<1,3,-2>; v=<-5,3,2>

2.u=<1,-2,4>; v=<5,3,7>

Answer 1

1.u=<1,3,-2>; v=<-5,3,2>

The vectors are

u = i + 3j - 2k

v = -5i +3j + 2k

$\cos \theta$ = $\dfrac{u.v}{|u||v|}$

cosθ = $\dfrac{(i + 3j - 2k ) . (-5i +3j + 2k) }{\sqrt{14}*\sqrt{38}}$

cosθ = -$\dfrac{- 5 + 9 - 4}{\sqrt{14}*\sqrt{38}}$ = 0

$\theta$ = $\dfrac{\pi}{2}$

Hence the vectors u and v are orthogonal to each other.

2.u=<1,-2,4>; v=<5,3,7>

The vectors are

u = i - 2j + 4k

v = 5i +3j + 7k

$\cos \theta$ = $\dfrac{u.v}{|u||v|}$

cosθ = $\dfrac{(i - 2j + 4k ) . (5i +3j + 7k) }{\sqrt{21}*\sqrt{83}}$

cosθ = $\dfrac{5 - 6 + 28}{\sqrt{21}*\sqrt{83}}$

$\cos \theta$ = $\dfrac{27}{\sqrt{21}*\sqrt{83}}$

$\cos\theta$ = 0.647

$\theta$ = $49.68\degree$

Since, $\theta$ < 90$\degree$ so u and v make the acute angle with each other.