Mathematics

Problem

For vectors

\mathbf{u}

and

\mathbf{v}

it is known that

\mathbf{u}\cdot\mathbf{v}=15

and

|\mathbf{u}|=5

. The angle between them is

45^{\circ}

. a. (i) Find the magnitude

|\mathbf{v}|

. b. (ii) If the angle between

\mathbf{u}

and

\mathbf{v}

were

135^{\circ}

instead of

45^{\circ}

while the magnitudes remained the same, what would be the new value of

\mathbf{u}\cdot\mathbf{v}

? c. (iii) Explain why the sign of the dot product changes when the angle is altered from

45^{\circ}

135^{\circ}