- #1

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- Homework Statement
- What is the one-parameter parametrization of a unit circle (with the center as the origin) with its axis spanned by the vector ##u## in ##\mathbb{R}^4##? What about the general one-variable parametrization of a unit circle in ##\mathbb{R}^n##?

- Relevant Equations
- ##\mathbb{SU}(2)##, ##\mathbb{S}^3##

I tried to looking at lower-dimensional cases:

For ##n=2## we have

$$(x(t),y(t))=(cos(t),sin(t))$$

For ##n=3## we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to

$$(x(t),y(t),z(t))=(cos(t)(a_1,a_2,a_3)+sin(t)(b_1,b_2,b_3))$$

It seems like there was a pattern for ##n=2## and ##n=3##. But, there is no reason to think the pattern continues for larger values of ##n##. This is a wild guess I think?

For ##n=4##, we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to

$$(x(t),y(t),z(t),w(t))=(cos(t)(a_1,a_2,a_3,a_4)+sin(t)(b_1,b_2,b_3,b_4))$$

For general ##n##, we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to

$$\vec{s}(t)=(cos(t)\vec{a}+sin(t)\vec{b})$$

For ##n=2## we have

$$(x(t),y(t))=(cos(t),sin(t))$$

For ##n=3## we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to

$$(x(t),y(t),z(t))=(cos(t)(a_1,a_2,a_3)+sin(t)(b_1,b_2,b_3))$$

It seems like there was a pattern for ##n=2## and ##n=3##. But, there is no reason to think the pattern continues for larger values of ##n##. This is a wild guess I think?

For ##n=4##, we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to

$$(x(t),y(t),z(t),w(t))=(cos(t)(a_1,a_2,a_3,a_4)+sin(t)(b_1,b_2,b_3,b_4))$$

For general ##n##, we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to

$$\vec{s}(t)=(cos(t)\vec{a}+sin(t)\vec{b})$$