A particle moves with a velocity v = 5î - 3ĵ + 6k̂ ms⁻¹ under the influence of a constant force F = 10î + 10ĵ + 20k̂ N. The instantaneous power applied to the particle is

The correct option is 140 Js⁻¹.

Understanding Instantaneous Power:
Instantaneous power (P) delivered by a force to a moving particle is defined as the rate at which work is done. Mathematically, it is given by the dot product (scalar product) of the force vector and the velocity vector:
$P=\overrightarrow{F}·\overrightarrow{v}$

Step-by-Step Calculation:
We are given the velocity vector and the force vector as follows:
$\overrightarrow{v}=5\hat{i}-3\hat{j}+6\hat{k}{\text{ms}}^{-1}$
$\overrightarrow{F}=10\hat{i}+10\hat{j}+20\hat{k}\text{N}$

Now, we compute the dot product by multiplying the corresponding components (x, y, and z) of both vectors:
$P=(F_x·v_x)+(F_y·v_y)+(F_z·v_z)$
Substitute the values into the equation:
$P=(10·5)+(10·(-3\left)\right)+(20·6)$
Simplify each term:
$P=50-30+120$
Calculate the final value:
$P=140{\text{Js}}^{-1}$

Thus, the instantaneous power applied to the particle is 140 Js⁻¹ (or Watts).