$($a$)$ Given the vectors $a=3i-j-2k$ and $b=4i-3j+5k,$ calculate $a*b.$

$($b$)$ Given the vectors $a=2i-j-4k$ and $b=i+j+2k,$ calculate $ab.$

$($c$)$ Let $P_1,P_2$ and $P_3$ be points relative to a fixed origin $O$ and let $r_1,r_2$ and $r_3$ be the respective position vectors from $O$ to each point. Suppose the vector equation:

$r_1+r_2+r_3=0$

holds with respect to $O.$ Show that it holds with respect to any other origin $O^{\text{'}}$ if and only if $++=0.$

$($d$)$ Given a parallelogram ABCD and the vectors a from $A$ to $D,b$ from $A$ to $B,c$ from $A$ to $C$ and $d$ from $B$ to $D.$ Show that $|cd|=2|ab|.$

$($e$)$ Determine the projection of the vector $a=3i+4j+k$ onto the vector $b=-4i+5j-3k.$

$2.($a$)$ A person drives from town $A$ to town $B$ and there are two roads available. The first road is of length $l$ and the driver can cruise at constant speed $v.$ The second road is of length $\frac{3}{4}l$ and a certain proportion $()$ of the road is under snow. Without snow on the road they can travel with speed $v$ but with snow they can only travel with speed $\frac{1}{2}v.$

$($i$)$ Write down and simplify expressions for the travel time on each road.

$($ii$)$ What is the value of $$ below which it is faster to travel on the second road?

$($b$)$ A small rocket, with a booster attached, blasts off and heads straight upward. At a height of $5$ km and velocity of $200m{s}^{-1},$ it releases the booster. You may neglect air resistance and assume the booster acts under the influence of acceleration due to gravity. Take $g=9\mathrm{.}81m{s}^{-2}.$

$($i$)$ What is the maximum height the booster attains?

$($ii$)$ What is the velocity of the booster at a height of $6$ km $?$ Explain the significance of the sign of your answer.

$($iii$)$ Suppose that as the booster travels back down through the height of $6$ km its rocket reignites for $1$ s imparting an additional upward acceleration $a.$ Determine the value of a above which the booster reverses its motion.

$3.($a$)$ A projectile is launched at an angle $$ with the horizontal and initial speed $v_0.$ Derive an expression for the trajectory of the projectile in terms of the horizontal $(x)$ and vertical $(z)$ components of displacement and $tan.$

$($b$)$ A particle is projected, at angle $$ to the horizontal and initial speed $v_0,$ from the origin. The particle just clears a wall of height $h$ a horizontal distance $h$ away. Given that only one such trajectory exists, show that $tan=\frac{v_0^2}{gh}.$

$($c$)$ Two gaelic football players kick a football imparting the same initial speed $v_0.$ However, the first player kicks at an angle $=\frac{}{3}$ with the horizontal and the second player kicks with $=\frac{}{6}.$ After kicking the ball, each player runs with constant speed in a straight line and just manages to catch the ball before it hits the ground.

$($i$)$ Write down the velocity vector $(v)$ of the ball during flight for arbitrary $.$

$($ii$)$ What is the time of flight $(T)$ of the ball for arbitrary $?$ Write down an expression for $v$ at this time and at time $t=0.$

$($iii$)$ Determine which player runs the fastest.