Question #208120

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>


1
Expert's answer
2021-06-18T08:40:33-0400

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



The vectors are


u = i + 3j - 2k

v = -5i +3j + 2k


cosθ\cos \theta = u.vuv\dfrac{u.v}{|u||v|}


cosθ = (i+3j2k).(5i+3j+2k)1438\dfrac{(i + 3j - 2k ) . (-5i +3j + 2k) }{\sqrt{14}*\sqrt{38}}


cosθ = -5+941438\dfrac{- 5 + 9 - 4}{\sqrt{14}*\sqrt{38}} = 0


θ\theta = π2\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


cosθ\cos \theta = u.vuv\dfrac{u.v}{|u||v|}


cosθ = (i2j+4k).(5i+3j+7k)2183\dfrac{(i - 2j + 4k ) . (5i +3j + 7k) }{\sqrt{21}*\sqrt{83}}


cosθ = 56+282183\dfrac{5 - 6 + 28}{\sqrt{21}*\sqrt{83}}


cosθ\cos \theta = 272183\dfrac{27}{\sqrt{21}*\sqrt{83}}


cosθ\cos\theta = 0.647


θ\theta = 49.68°49.68\degree



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





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