Question

# 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)

Vectors and spaces
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 $$\displaystyle{90}^{{\circ}}$$. u = (-1, -1, 8, 0), v = (5,6,1,4)

2020-10-22
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
$$\displaystyle{d}{\left({u},{v}\right)}={\left|{{\left|{{u}-{v}}\right|}}\right|}=\sqrt{{{\left(-{6}\right)}^{{2}}+{\left(-{7}\right)}^{{2}}+{7}^{{2}}+{\left(-{4}\right)}^{{2}}}}=\sqrt{{{150}}}={5}\sqrt{{{6}}}$$
Furthermore, the angle 0 between these vectors is given by
$$\displaystyle{\cos{{0}}}=\frac{{\left\langle{u},{v}\right\rangle}}{{\left|{{\left|{{u}-{v}}\right|}}\right|}}$$,
where $$\displaystyle{\left\langle{u},{v}\right\rangle}$$ is a scalar product of u and v. So we compute
$$\displaystyle{\left\langle{u},{v}\right\rangle}=-{5}-{6}+{8}+{0}=-{3}$$
$$\displaystyle{\left|{{\left|{{u}}\right|}}\right|}=\sqrt{{{\left(-{1}\right)}^{{2}}+{\left(-{1}\right)}^{{2}}+{8}^{{2}}+{0}^{{2}}}}=\sqrt{{{66}}}$$
$$\displaystyle{\left|{{\left|{{v}}\right|}}\right|}=\sqrt{{{5}^{{2}}+{6}^{{2}}+{1}^{{2}}+{4}^{{2}}}}=\sqrt{{{78}}}$$
Therefore,
$$\displaystyle{\cos{{0}}}=\frac{{-{3}}}{{\sqrt{{{66}}}\sqrt{{{78}}}}}$$
which means that
$$\displaystyle{0}=\frac{{\arccos{{\left(-{3}\right)}}}}{{\sqrt{{{66}}}\sqrt{{{78}}}}}\approx{1.613}$$
So, this angle is obtuse.
Result
Distance:
$$\displaystyle{d}{\left({u},{v}\right)}={5}\sqrt{{{6}}}$$
Angle:
$$\displaystyle{0}=\frac{{\arccos{{\left(-{3}\right)}}}}{{\sqrt{{{66}}}\sqrt{{{78}}}}}\approx{1.613}$$
It is obtuse.