General true of false questions about vector function in calc 3

[zhuyilun]

Homework Statement

a. the derivative of a vector function is obtained by differentiating each component function
b. if r(t) is a differentiable vector function, then d/dt the magnitude of r(t) = the magnitude of r'(t)
c. the binormal vector is B(t) =N(t)xT(t)
d. if k(t)=0 for all t, the curve is a straight line
e. if the magnitude of r(t)=1, then r'(t) is orthogonal to r(t) for all t
f. different parametrizations of the same curve result in identical tangent vectors at a given point on the curve

Homework Equations

The Attempt at a Solution

i think:
a is T
b is F
c i have no idea about what binormal vector is. is it a vector that is orthogonal to both two vectors? if so the cross product would give a vector that is orthogonal to both vectors. so c would be T, i am not quite sure about this one
d i have no idea
e i have no idea
f i think it's T, because although its different parametrizations, the curve is still the same . therefore, the tangent lines at a given point are the same

 

[Mark44]

Homework Statement

a. the derivative of a vector function is obtained by differentiating each component function
b. if r(t) is a differentiable vector function, then d/dt the magnitude of r(t) = the magnitude of r'(t)
c. the binormal vector is B(t) =N(t)xT(t)
d. if k(t)=0 for all t, the curve is a straight line
e. if the magnitude of r(t)=1, then r'(t) is orthogonal to r(t) for all t
f. different parametrizations of the same curve result in identical tangent vectors at a given point on the curve

Homework Equations

The Attempt at a Solution

i think:
a is T
b is F
c i have no idea about what binormal vector is. is it a vector that is orthogonal to both two vectors? if so the cross product would give a vector that is orthogonal to both vectors. so c would be T, i am not quite sure about this one

Yes, the binormal B is perpendicular to (normal to) the other two vectors.

d i have no idea

What does k(t) represent? Isn't it the curvature? If so, what does it mean to say that k(t) = 0 for all t?

e i have no idea

If |r(t)| = 1, what sort of curve do you have?

f i think it's T, because although its different parametrizations, the curve is still the same . therefore, the tangent lines at a given point are the same

 

[zhuyilun]

Yes, the binormal B is perpendicular to (normal to) the other two vectors.

What does k(t) represent? Isn't it the curvature? If so, what does it mean to say that k(t) = 0 for all t?

k(t) is the curvature, therefore, i got T'(t)=o, but what does that tell me?

If |r(t)| = 1, what sort of curve do you have?

i think i got a circle/sphere, therefore, does that necessarily mean r'(t) is orthogonal to r(t)?

btw, is my answer to the last question right?
thank you