VC1.10

Let r be the position vector of any point P(x,y,z) on the given curve at time t.Then, we have

$r=xi+yj+zk=(t^3+1)i+t^2 j+(2t+5)k\,$

Therefore, velocity= $\frac{dr}{dt}=3t^2 i+2tj+2k\,$

Acceleration= $\frac{d^2 r}{dt^2}=6ti+2j\,$

At t=1, $v=3i+2j+2k,a=6i+2j\,$

Unit vector in the direction of $i+j+3k\,$ is $\frac{i+j+3k}{\sqrt{11}}\,$

Therefore, Required component of velocity is the dot product of the velocity vector and the unit vector above.

$(3i+2j+2k)\cdot (\frac{i+j+3k}{\sqrt{11}})=\frac{11}{\sqrt{11}}=\sqrt{11}\,$

Similarly,Required component of acceleration is the dot product of the acceleration vector and the unit vector above.

$(6i+2j)\cdot (\frac{i+j+3k}{\sqrt{11}})=\frac{8}{\sqrt{11}}\,$