Question

If \(\vec F(t)\) has constant direction, then

• A. \(\vec F\cdot\dfrac{d\vec F}{dt}=0\)
• B. \(\vec F\times\dfrac{d\vec F}{dt}=0\)
• C. \(\vec F\cdot\dfrac{d\vec F}{dt}=c\)
• D. \(\vec F\times\dfrac{d\vec F}{dt}=c\)

Hint:

Vectors with the same direction are parallel, and the cross product of parallel vectors is zero.

Answer 1

The Correct Option is B

Concept:
If a vector has constant direction, then only its magnitude may change with time. This means \(\vec F(t)\) and its derivative \(\dfrac{d\vec F}{dt}\) are parallel vectors. The cross product of two parallel vectors is zero.

Step 1: Represent a vector with constant direction. If direction is constant, we can write \[ \vec F(t)=\lambda(t)\vec a \] where \(\vec a\) is a constant vector and \(\lambda(t)\) is a scalar function.

Step 2: Differentiate with respect to \(t\). \[ \frac{d\vec F}{dt}=\lambda'(t)\vec a \] So both \[ \vec F(t)=\lambda(t)\vec a \] and \[ \frac{d\vec F}{dt}=\lambda'(t)\vec a \] are in the same direction.

Step 3: Use cross product property. If two vectors are parallel, then their cross product is zero. Hence, \[ \vec F\times \frac{d\vec F}{dt}=0 \]

Step 4: Final answer. \[ \boxed{\vec F\times\dfrac{d\vec F}{dt}=0} \]