Solved: "Find the Euclidean distance between u and v..."

Brennan Flores

Answered question

2020-10-21

Find the Euclidean distance between u and v and the cosine of the angle between those vectors. State whether that angle is acute, obtuse, or

90^{\circ}

. u = (-1, -1, 8, 0), v = (5,6,1,4)

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Skilled2020-10-22Added 108 answers

The Eunclidean distance between u and v is the Euclidean norm of the vector u - v. Thus, we must find
u-v=(-1,-1,8,0)-(5,6,1,4)=(-6,-7,7,-4)
and

d (u, v) = | | u - v | | = \sqrt{{(- 6)}^{2} + {(- 7)}^{2} + 7^{2} + {(- 4)}^{2}} = \sqrt{150} = 5 \sqrt{6}

Furthermore, the angle 0 between these vectors is given by

\cos 0 = \frac{⟨ u, v ⟩}{| | u - v | |}

,
where

⟨ u, v ⟩

is a scalar product of u and v. So we compute

⟨ u, v ⟩ = - 5 - 6 + 8 + 0 = - 3

| | u | | = \sqrt{{(- 1)}^{2} + {(- 1)}^{2} + 8^{2} + 0^{2}} = \sqrt{66}

| | v | | = \sqrt{5^{2} + 6^{2} + 1^{2} + 4^{2}} = \sqrt{78}

Therefore,

\cos 0 = \frac{- 3}{\sqrt{66} \sqrt{78}}

which means that

0 = \frac{\arccos (- 3)}{\sqrt{66} \sqrt{78}} \approx 1.613

So, this angle is obtuse.
Result
Distance:

d (u, v) = 5 \sqrt{6}

Angle:

0 = \frac{\arccos (- 3)}{\sqrt{66} \sqrt{78}} \approx 1.613

It is obtuse.

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