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

midtlinjeg 2021-10-30 Answered
At what points does the helix \(\displaystyle{r}{\left({t}\right)}={<}{\sin{{t}}},{\cos{{t}}},{t}{>}\)intersect the sphere \(\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={5}\)

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Expert Answer

Maciej Morrow
Answered 2021-10-31 Author has 4398 answers
Step 1
We have to find points where helix \(\displaystyle{r}{\left({x}\right)}={\left\langle{\sin{{t}}},{\cos{{t}}}\right\rangle}\) intersects the sphere \(\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={5}\)
Take each vector component from the given vector and plug it into the sphere's equation (\(\displaystyle{x}=\text{vector's }\ {x}-\text{component},{y}=\text{vector's }\ ,{y}-\text{component},{z}=\text{vector's }\ ,{z}-\text{component}\))
In this way, we get,
\(\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={5}\)
\(\displaystyle{{\sin}^{{2}}{t}}+{{\cos}^{{2}}{t}}+{t}^{{2}}={5}\)
\(\displaystyle{\left({{\sin}^{{2}}{t}}+{{\cos}^{{2}}{t}}\right)}+{t}^{{2}}={5}\ \ \ \ {\left(\text{apply rule }\ {{\sin}^{{2}}\theta}+{{\cos}^{{2}}\theta}={1}\right)}\)
\(\displaystyle{1}+{t}^{{2}}={5}\)
\(\displaystyle{1}+{t}^{{2}}-{1}={5}-{1}\)
\(\displaystyle{t}^{{2}}={4}\)
\(\displaystyle{t}^{{2}}={\left(\pm{2}\right)}^{{2}}\)
\(\displaystyle{t}=\pm{2}\)
Step 2
\(\displaystyle\text{Using the values for t, plug in t for each of the vector's components.}\)
\(\displaystyle\text{Due to the }\ \pm\ \text{ in the solution t, there will be two sets of intersecting points.}\)
For t=2, we have
Set 1:\(\displaystyle{<}{\sin{{\left({2}\right)}}},{\cos{{\left({2}\right)}}},{2}{>}\)
Set 1:\(\displaystyle{\left({0.909},{0.416},{2}\right)}\)
For t=-2, we have,
Set 2:\(\displaystyle{<}{\sin{{\left(-{2}\right)}}},{\cos{{\left(-{2}\right)}}},-{2}{>}\)
Set 2:\(\displaystyle{\left(-{0.909},-{0.416},-{2}\right)}\)
Result
Set 1:\(\displaystyle{\left({0.909},{0.416},{2}\right)}\)
Set 2:\(\displaystyle{\left(-{0.909},-{0.416},-{2}\right)}\)
These two sets represent the intersection points of helix \(\displaystyle{r}{\left({x}\right)}={\left\langle{\sin{{t}}},{\cos{{t}}},{t}\right\rangle}\) and sphere
\(\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={1}\)
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