At what points does the helix r(t)=&lt;sin⁡t,cos⁡t,t&gt;intersect the sphere x2+y2+z2=5

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At what points does the helix r(t)=&amp;lt;sin⁡t,cos⁡t,t&amp;gt;intersect the sphere x2+y2+z2=5

Maciej Morrow · Accepted Answer

Step 1We have to find points where helix r(x)=⟨sin⁡t,cos⁡t⟩  intersects the sphere  x2+y2+z2=5Take each vector component from the given vector and plug it into the sphere&#039;s equation (x=vector&#039;s  x−component,y=vector&#039;s  ,y−component,z=vector&#039;s  ,z−component)In this way, we get,x2+y2+z2=5sin2t+cos2t+t2=5(sin2t+cos2t)+t2=5    (apply rule  sin2θ+cos2θ=1)1+t2=51+t2−1=5−1t2=4t2=(±2)2t=±2Step 2Using the values for t, plug in t  for each of the vector&#039;s components.Due to the  ±  in the solution t, there will be two sets of intersecting points.For t=2, we haveSet 1:&amp;lt;sin⁡(2),cos⁡(2),2&amp;gt;Set 1:(0.909,0.416,2)For t=-2, we have,Set 2:&amp;lt;sin⁡(−2),cos⁡(−2),−2&amp;gt;Set 2:(−0.909,−0.416,−2)ResultSet 1:(0.909,0.416,2)Set 2:(−0.909,−0.416,−2)These two sets represent the intersection points of helix r(x)=⟨sin⁡t,cos⁡t,t⟩ and sphere x2+y2+z2=1

At what points does the helix r(t)=<\sin t, \cos t, t> intersect the spher

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