Differentiation of vector function(explanation of solution)

[negation]

Homework Statement

Show that if the vector function r(t) is continuously differentiable for t on an interval I and |r(t)| = c, a constant for all t \in I, then r'(t) is orthorgonal to r(t) for all t \in I

What would the curve described by r(t) look like?

The Attempt at a Solution

The solution is given as:

|r(t)|² = r(t) . r(t)

r(t).r(t) = c² ( I can see the algebraic reasoning for this but what's the conceptual reasoning behind?)

then,

r'(t).r(t) + r(t).r'(t) = 0

that is, 2r'(t) .r(t) = 0. Hence r'(t) is orthorgonal to r(t) for all t.

What is the question I should be asking myself?

 

[Curious3141]

Homework Statement

Show that if the vector function r(t) is continuously differentiable for t on an interval I and |r(t)| = c, a constant for all t \in I, then r'(t) is orthorgonal to r(t) for all t \in I

What would the curve described by r(t) look like?

The Attempt at a Solution

The solution is given as:

|r(t)|² = r(t) . r(t)

r(t).r(t) = c² ( I can see the algebraic reasoning for this but what's the conceptual reasoning behind?)

then,

r'(t).r(t) + r(t).r'(t) = 0

that is, 2r'(t) .r(t) = 0. Hence r'(t) is orthorgonal to r(t) for all t.

What is the question I should be asking myself?

Not one, but three questions that I would suggest:

1) What is the definition of a dot product? Don't just quote a formula, define the terms.

2) What is the dot product of a non-zero vector with itself? Go back to the definition in 1) to answer this.

3) What is the dot product of two non-zero vectors that are perpendicular to each other? Conversely, what can be said when two non-zero vectors have a dot product of zero?

I assume you get the product rule part (works the same as for scalar variables) and the commutativity and associativity parts allowing you to group the terms. If not, you should clarify exactly what doubts you have about those.

 

[negation]

Not one, but three questions that I would suggest:

1) What is the definition of a dot product? Don't just quote a formula, define the terms.

2) What is the dot product of a non-zero vector with itself? Go back to the definition in 1) to answer this.

3) What is the dot product of two non-zero vectors that are perpendicular to each other? Conversely, what can be said when two non-zero vectors have a dot product of zero?

I assume you get the product rule part (works the same as for scalar variables) and the commutativity and associativity parts allowing you to group the terms. If not, you should clarify exactly what doubts you have about those.

I suppose it's about understanding the question. The solutions appear pretty trivial but I'm having a slightly difficult time interpreting what the question wants of me exactly.

 

[Mark44]

I suppose it's about understanding the question. The solutions appear pretty trivial but I'm having a slightly difficult time interpreting what the question wants of me exactly.

Show that if the vector function r(t) is continuously differentiable for t on an interval I and |r(t)| = c, a constant for all t \in I, then r'(t) is orthorgonal to r(t) for all t \in I

Show that r'(t) is orthogonal to r(t) for all t in the interval. "Orthogonal to" should suggest something to you.

What would the curve described by r(t) look like?

You're given that |r(t)| is constant. Do you have any ideas about what such a curve might look like?

 

[negation]

Show that r'(t) is orthogonal to r(t) for all t in the interval. "Orthogonal to" should suggest something to you.
You're given that |r(t)| is constant. Do you have any ideas about what such a curve might look like?

It suggest that for the interval for which t is defined, r'(t) dot r(t) = 0.

I don't. Wolfram was unable to plot the graph too. But, I'm going in on a leg and state that it is an absolute value graph? y = |r(t)|?

 

[PeroK]

It suggest that for the interval for which t is defined, r'(t) dot r(t) = 0.

I don't. Wolfram was unable to plot the graph too. But, I'm going in on a leg and state that it is an absolute value graph? y = |r(t)|?

I think you're going round in circles on this problem!

 

[Mark44]

i think you're going round in circles on this problem!

lol!

 

[Mark44]

It suggest that for the interval for which t is defined, r'(t) dot r(t) = 0.

This is a good start. Can you rewrite the dot product using its definition?

I don't. Wolfram was unable to plot the graph too. But, I'm going in on a leg and state that it is an absolute value graph? y = |r(t)|?