Vector Valued Functions

Question

Let r(t)= <sin(10t),cos(10t),sin(10t)cos(20t)>

Find the point where r(t) intersects the xy-plane on the interval pi/10 < x < 3pi/20

Answer 1

Hi Amal A.,

If I'm not mistaken, your function of t has (x,y,z) dimensions? Then solve once for z=0 (either sin(10t) or cos(20t) or both are zero), and then examine those (t) points for the appropriate range for x (sin 10t is in the stated range).

Shouldn't be hard? In fact, you can drop the sin(10t)=0 solutions, since those don't meet the range ipse facto..

--Cheers, -- Mr. d.

Answer 2

The point where r(t) intersects the xy-plane means the z component is zero. So let's set it to zero and we have:

sin(10t)cos(20t) = 0

To get zero value, either one of them can be zero:

sin(10t) = 0 or cos(20t) = 0

Sine function is zero when the following condition is satisfied:

10t = n*pi, where n = 0,1,2,...

Meanwhile cosine function is zero when:

20t = m*pi/2, where m = 1,3,5,7,...

The rest should be straightforward.