- #1
Are a vector and its derivative perpendicular at all times?
- Context: Undergrad
- Thread starter TheCanadian
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SUMMARY
The discussion clarifies that a vector and its derivative are not always perpendicular. Specifically, when a vector's magnitude changes while its direction remains constant, the derivative points in the same direction as the vector. This contrasts with scenarios where the vector's magnitude is constant, such as in uniform circular motion, where the derivative is indeed perpendicular to the vector. The conversation highlights the importance of understanding both normal and tangent components of a vector's derivative.
PREREQUISITES- Understanding of vector calculus
- Familiarity with derivatives in physics
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- Learn about uniform circular motion and its mathematical implications
- Investigate normal and tangent components of vectors in detail
Students and professionals in physics, mathematics, and engineering who seek to deepen their understanding of vector behavior and derivatives in various motion scenarios.
- #2
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No. What if the vector is not changing direction? In general, the derivative of a vector has components both normal and tangent to the vector.
Chet
Chet
- #3
Nugatory
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TheCanadian said:
No. For an obvious example, consider a vector whose magnitude but not direction is increasing as a function of time: ##\vec{F}(t+\Delta{t})-\vec{F}(t)## points in the same direction as ##\vec{F}(t)##. You're thinking of the case in which the magnitude of the vector is constant over time, in which case the derivative must indeed be perpendicular (as in acceleration in the case of uniform circular motion).
[Edit: Chet got there first but I used more Latex so I still win
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