Mathematics

Problem

A vector

\mathbf{p}=(t,2,5)

depends on the real parameter

t

. Another vector is fixed as

\mathbf{q}=(3,-1,4)

. a. (i) Show that the dot product

\mathbf{p}\!\cdot\!\mathbf{q}

can be expressed as

3t-2+20

. b. (ii) Determine the value(s) of

t

for which the angle between

\mathbf{p}

and

\mathbf{q}

120^{\circ}

. c. (iii) For each value of

t

obtained, verify that the corresponding angle is indeed obtuse without directly using a trigonometric table.