At what times does the object intersect one of the coordinate planes?

[ShakeECET109]

Homework Statement

r(t)={sin(pi*t),ln(t),((1/4)e^t}

At what time(s) does the object intersect one of the coordinate axes?

At what time(s) does the object intersect one of the coordinate planes?

During what times t is the object in the first octant?

Homework Equations

Not sure??

The Attempt at a Solution

I know points on the z-axis have the form (0,0,z). The curve will intersect the z-axis if x(t) and y(t) are both 0 at the same time (t value). Same for the x-axis and y-axis.

Points in the yz-plane have the form (0,y,z), the x-coordinate of the point must be 0. So for this I need to find the values of t for which x(t)=0. Same for the xy- and xz-planes.

I really am not sure how to start. Can anyone point me in the right direction?? Formula maybe?? Or show me an example done using a position vector?

 

[Mark44]

Homework Statement

r(t)={sin(pi*t),ln(t),((1/4)e^t}

At what time(s) does the object intersect one of the coordinate axes?

At what time(s) does the object intersect one of the coordinate planes?

During what times t is the object in the first octant?

Homework Equations

Not sure??

The Attempt at a Solution

I know points on the z-axis have the form (0,0,z). The curve will intersect the z-axis if x(t) and y(t) are both 0 at the same time (t value). Same for the x-axis and y-axis.

Points in the yz-plane have the form (0,y,z), the x-coordinate of the point must be 0. So for this I need to find the values of t for which x(t)=0. Same for the xy- and xz-planes.

I really am not sure how to start. Can anyone point me in the right direction?? Formula maybe?? Or show me an example done using a position vector?

Your reasoning is correct, so you have already made a good start. Your vector function r(t) lists x(t), y(t), and z(t). For the first problem you need to have sin(pi*t) = 0 and ln(t) = 0 for the same value of t.

A point is in the first octant if all three coordinates are positive.

 

[ShakeECET109]

Your reasoning is correct, so you have already made a good start. Your vector function r(t) lists x(t), y(t), and z(t). For the first problem you need to have sin(pi*t) = 0 and ln(t) = 0 for the same value of t.

A point is in the first octant if all three coordinates are positive.

sin(pi*t) = 0
sooo sin(pi*t)=ln(t) , then solve for t?
ln(t) = 0

 

[Mark44]

sin(pi*t) = 0
sooo sin(pi*t)=ln(t) , then solve for t?
ln(t) = 0

No, you don't just want the points at which sin(pi*t) = ln(t) - you want to solve the system of equations
sin(pi*t) = 0
ln(t) = 0

 

[ShakeECET109]

Ohhh soo...
[URL]http://www2.wolframalpha.com/Calculate/MSP/MSP118919ha265g6c4b58g80000500gabba95c6h992?MSPStoreType=image/gif&s=23&w=156&h=18[/URL]
[URL]http://www2.wolframalpha.com/Calculate/MSP/MSP119419ha265g6c4b58g8000022fg02efc8adeba9?MSPStoreType=image/gif&s=23&w=500&h=22[/URL]

so x and y are 0 when t = 1 and I have to do the same for the x and y-axis because that was for the z axis?

 

[ShakeECET109]

any help on how to find when it intersects a plane??

 

[Mark44]

Your image links are bad.

Ohhh soo...
[URL]http://www2.wolframalpha.com/Calculate/MSP/MSP118919ha265g6c4b58g80000500gabba95c6h992?MSPStoreType=image/gif&s=23&w=156&h=18[/URL]
[URL]http://www2.wolframalpha.com/Calculate/MSP/MSP119419ha265g6c4b58g8000022fg02efc8adeba9?MSPStoreType=image/gif&s=23&w=500&h=22[/URL]

so x and y are 0 when t = 1 and I have to do the same for the x and y-axis because that was for the z axis?

Yes.

 

[Mark44]

any help on how to find when it intersects a plane??

Do what you described in post #1

Points in the yz-plane have the form (0,y,z), the x-coordinate of the point must be 0. So for this I need to find the values of t for which x(t)=0. Same for the xy- and xz-planes.