VC1.10

From Example Problems
Jump to: navigation, search

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}}}}\,

Main Page