A particle has an angular speed of 3rad/s and the axis of rotation passes through the points (1,1,2) and (1,2,−2). Find the velocity of the particle at point P(3,6,4).

To solve this problem, we need to find the vector of the axis of rotation, the position vector of the point P, and then calculate the velocity using the formula for the velocity of a particle in rotational motion.

Step 1: Find the vector of the axis of rotation The axis of rotation passes through the points (1,1,2) and (1,2,-2). The vector of this axis, denoted as r, can be found by subtracting the coordinates of the first point from the second point.

r = (1-1, 2-1, -2-2) = (0, 1, -4)

Step 2: Find the position vector of the point P The position vector of the point P(3,6,4), denoted as p, is simply its coordinates.

p = (3, 6, 4)

Step 3: Find the vector from the axis of rotation to the point P This vector, denoted as d, is the difference between the position vector of the point P and the vector of the axis of rotation.

d = p - r = (3-0, 6-1, 4-(-4)) = (3, 5, 8)

Step 4: Calculate the velocity of the particle The velocity of a particle in rotational motion, denoted as v, is given by the cross product of the angular speed and the vector from the axis of rotation to the point, multiplied by the angular speed. The angular speed is given as 3 rad/s.

v = 3 * (r x d)

Calculating the cross product, we get:

r x d = (1*(-4) - 50, -43 - 00, 05 - 1*3) = (-4, -12, -3)

Multiplying by the angular speed, we get:

v = 3 * (-4, -12, -3) = (-12, -36, -9)

So, the velocity of the particle at point P(3,6,4) is (-12, -36, -9) m/s.