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Calculus Level 3

Given that the vector function r ( t ) r(t) has constant magnitude. What is the value of r ( t ) ⋅ r ′ ( t ) r(t)\cdot r'(t)

It is a linear function with non zero slope 2 There isn't enough information 1 0

Hamza A by Hamza A , 19, USA

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1 solution

Otto Bretscher
Nov 5, 2018

Use the product rule: 0 = ( ∣ ∣ r ( t ) ∣ ∣ 2 ) ′ = ( r ( t ) ⋅ r ( t ) ) ′ = 2 ( r ( t ) ⋅ r ′ ( t ) ) 0=(||r(t)||^2)'=(r(t)\cdot r(t))'=2(r(t) \cdot r'(t)) so r ( t ) ⋅ r ′ ( t ) = 0 r(t) \cdot r'(t)=\boxed{0}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

For a Three Dimensions, we can visualize the vector function as a radius of a sphere and thus the derivative will be the tangent vector which is ALWAYS perpendicular................

Aaghaz Mahajan - 2 years, 7 months ago

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Exactly! But it is good to have a simple proof in higher dimensions as well...

Otto Bretscher - 2 years, 7 months ago

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