 # 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) Brennan Flores 2020-10-21 Answered
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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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
$d\left(u,v\right)=||u-v||=\sqrt{{\left(-6\right)}^{2}+{\left(-7\right)}^{2}+{7}^{2}+{\left(-4\right)}^{2}}=\sqrt{150}=5\sqrt{6}$
Furthermore, the angle 0 between these vectors is given by
$\mathrm{cos}0=\frac{⟨u,v⟩}{||u-v||}$,
where $⟨u,v⟩$ is a scalar product of u and v. So we compute
$⟨u,v⟩=-5-6+8+0=-3$
$||u||=\sqrt{{\left(-1\right)}^{2}+{\left(-1\right)}^{2}+{8}^{2}+{0}^{2}}=\sqrt{66}$
$||v||=\sqrt{{5}^{2}+{6}^{2}+{1}^{2}+{4}^{2}}=\sqrt{78}$
Therefore,
$\mathrm{cos}0=\frac{-3}{\sqrt{66}\sqrt{78}}$
which means that
$0=\frac{\mathrm{arccos}\left(-3\right)}{\sqrt{66}\sqrt{78}}\approx 1.613$
So, this angle is obtuse.
Result
Distance:
$d\left(u,v\right)=5\sqrt{6}$
Angle:
$0=\frac{\mathrm{arccos}\left(-3\right)}{\sqrt{66}\sqrt{78}}\approx 1.613$
It is obtuse.