Given:-
[tex]<1,5>,<3,15>[/tex]
To find:-
The given vectors are parallel, orthogonal or neither.
So now we check, the given vectors is orthogonal or not.
[tex]\begin{gathered} \bar{u}\bar{.v}=1(3)+5(15) \\ \text{ =3+75} \\ \text{ =78} \end{gathered}[/tex]
The given vectors are not orthogonal because if the vectors are orthogonal the dot product should be zero.
So now we check, the given vectors is parallel or not,
[tex]\begin{gathered} \lvert u\rvert=\sqrt[]{1^2+5^2} \\ \text{ =}\sqrt[]{1+25} \\ \text{ =}\sqrt[]{26} \end{gathered}[/tex]
Also,
[tex]\begin{gathered} \lvert v\rvert=\sqrt[]{3^2+15^2} \\ \text{ =}\sqrt[]{9+225} \\ \text{ =}\sqrt[]{234} \end{gathered}[/tex]
So now,
[tex]\begin{gathered} \theta=\cos ^{-1}(\frac{u.v}{\lvert u\rvert\lvert v\rvert}) \\ \theta=\cos ^{-1}(\frac{78}{\sqrt[]{26}\sqrt[]{234}}) \\ \theta=\cos ^{-1}(\frac{78}{5.099\times15.29}) \\ \theta=\cos ^{-1}(1) \\ \theta=90 \end{gathered}[/tex]
So we get the value of theta as 90 degree. So the given vectors are Parallel.
