The angular velocity of a body is ω = 2î + 3ĵ + 4k̂ and a torque 𝜏 = î + 2ĵ + 3k̂ acts on it. The rotational power will be

To find the rotational power of the body, we can use the relationship between rotational power, torque, and angular velocity.

Rotational power ($P$) is defined as the dot product of the torque vector ($\tau$) and the angular velocity vector ($\omega$):

$P=\tau ·\omega$

We are given the following vector quantities:

Angular velocity:

$\omega =2\hat{i}+3\hat{j}+4\hat{k}$

Torque:

$\tau =\hat{i}+2\hat{j}+3\hat{k}$

Now, we compute the dot product of these two vectors:

$P=(\hat{i}+2\hat{j}+3\hat{k})·(2\hat{i}+3\hat{j}+4\hat{k})$

Multiplying the corresponding components:

$P=(1\times 2)+(2\times 3)+(3\times 4)$

$P=2+6+12$

$P=20\text{W}$

Thus, the rotational power is 20 W.