If the position vector of a point is given by r = (2t, 3t, 4t), what is the velo

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Question: If the position vector of a point is given by r = (2t, 3t, 4t), what is the velocity vector?

Options:

Correct Answer: (2, 3, 4)

Solution:

Velocity vector = dr/dt = (2, 3, 4)

If the position vector of a point is given by r = (2t, 3t, 4t), what is the velocity vector?

If the position vector of a point is given by r = (2t, 3t, 4t), what is the velocity vector?

Step 1: Understand that the position vector r is given as r = (2t, 3t, 4t). This means the position of the point changes with time t.
Step 2: To find the velocity vector, we need to differentiate the position vector r with respect to time t. This means we will find the derivative of each component of r.
Step 3: Differentiate the first component 2t with respect to t. The derivative is 2.
Step 4: Differentiate the second component 3t with respect to t. The derivative is 3.
Step 5: Differentiate the third component 4t with respect to t. The derivative is 4.
Step 6: Combine the derivatives from Steps 3, 4, and 5 to form the velocity vector. The velocity vector is (2, 3, 4).

Position Vector – The position vector r = (2t, 3t, 4t) represents the coordinates of a point in space as a function of time t.
Velocity Vector – The velocity vector is the derivative of the position vector with respect to time, indicating the rate of change of position.
Differentiation – The process of finding the derivative of a function, which in this case is applied to the position vector to find the velocity.