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)

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

Answers (1)

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.
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