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I am looking for the n-dimensional versions of formulas for the questions below:

• Find equation of plane that passes through 3 arbitrary points of $\mathbb{R}$$^3$ that have 3 coordinates

• Find vector equation of line that goes through two points in $\mathbb{R}$$^3$ that each have 3 coordinates

• Find the symmetric equations of a line through the point (a,b,c) and (e,f,g) in the same direction as some arbitrary vector in $\mathbb{R}$$^3$

• Find the parametric equations of the line that passes through a point in $\mathbb{R^{3}}$ and is parallel to the vector in $\mathbb{R}$$^3$

• Find the scalar equation of the line through two points in some arbitrary direction defined by a vector in $\mathbb{R}$$^3$

• Find the scalar equation of the line through two points.

• Find the equation of the tangent plane and symmetric equations of the normal line to the surface $A(x-a)^2+B(y-b)^2+C(x-c)^2=J$ at the point (q,r,t).

• Determine the equation of the line that passes through a point in $\mathbb{R}$$^3$ and normal to the plane ax+by-cz=k.

• Determine a normal vector and equation of the tangent plane to any surface with equation z= some polynomial at the point (a,b,c) in $\mathbb{R}$$^3$.

• Find the direction angle of some vector a$\vec i$+b$\vec j$+c$\vec > k$.

• Find the projection of some vector onto another vector.

  • Find the potential function for del f = some vector function ai +bi+ck.

  • Find the equation of the tangent plane to some plane at some point in $\mathbb{R}$$^2$

  • Find the gradient, directional derivative and equation of some tangent plane of some function with three variables at some point in $\mathbb{R}$$^3$ in some arbitrary direction.

  • Find the tangent plane and normal line to some surface at some point in $\mathbb{R}$$^3$.

I have a really hard time understanding how to compute the bulleted items above. Many of the stackexchange questions/online questions ask us for a specific plane with numbers. I'm more interested and think it would be morehelpful if I found out the general formula. I'm wondering whether you can extend my bulleted questions into $\mathbb{R^{n}}$ and points in $\mathbb{R^{n}}$. I think this would help me understand vectors in $\mathbb{R^{3}}$.

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    What exactly are you looking for in an answer? Do you just want formulas for each of those questions? If so, Googling them one by one should provide that. –  Jul 07 '17 at 13:45
  • I'm looking for formulas in $\mathbb{R}^n$ which google lacks. –  Jul 07 '17 at 13:45
  • I imagine you could find the formulas in $\Bbb R^n$ as well. But even if you can't, in many cases it's clear how to generalize the formula from $\Bbb R^3$ to $\Bbb R^n$. For instance, the formula for the first bullet point is $$\mathbf r(s,t) = \overrightarrow{OP}+s\overrightarrow{PQ} + t\overrightarrow{PR}$$ where $P,Q,R$ are your three points and $O$ is the origin. This formula actually works in $\Bbb R^n$, as well, but if you wanted to generalize it to a formula for higher dimensional flats in $\Bbb R^n$, you'd do it in the obvious way: –  Jul 07 '17 at 13:49
  • $$\mathbf r(t_1,\dots, t_{k-1}) = \overrightarrow{OP_1}+t_1\overrightarrow{P_1P_2} + \cdots + t_{k-1}\overrightarrow{P_1P_k}$$ where $P_1, \dots, P_k$ are the $k$ affinely independent points which define your $(k-1)$-flat. –  Jul 07 '17 at 13:49
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    These are not bad questions. But, how about one question at a time? Have you tried any one of them? Where did you get stuck? What difficulties do you have than there-are-too-many-questions-I-just-don't-have-time-to-finish-all-of-them-now? There is no silver bullet. –  Jul 07 '17 at 14:13
  • Do I need to know how to do everything in the bullets to study, curvature, curl, divergence, stoke's, green's theorem, vector fields, vector functions, cylindrical coordinates, line integrals, potential functions? –  Jul 08 '17 at 01:37

1 Answers1

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$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}$Rather than answer point by point, here are some useful generalities:

  • If $\Vec{v}$ and $\Vec{w}$ are non-zero vectors, the angle $\theta$ between then satisfies $$ \cos\theta = \frac{\Vec{v} \cdot \Vec{w}}{\|\Vec{v}\|\, \|\Vec{w}\|}. $$ Particularly, $\Vec{v}$ and $\Vec{w}$ are orthogonal if $\Vec{v} \cdot \Vec{w} = 0$.

  • If $\Vec{a}$ is a non-zero vector, then for every $\Vec{v}$, the components of $\Vec{v}$ parallel to $\Vec{a}$ and orthogonal to $\Vec{a}$ are $$ \operatorname{proj}_{\Vec{a}} \Vec{v} = \left(\frac{\Vec{v} \cdot \Vec{a}}{\|\Vec{a}\|^{2}}\right) \Vec{a},\qquad \operatorname{perp}_{\Vec{a}} \Vec{v} = \Vec{v} - \operatorname{proj}_{\Vec{a}} \Vec{v}. $$

  • If $\Vec{p}$ is a point and $\Vec{v}$ is a non-zero vector, then the set of points $\Vec{p} + t\Vec{v}$ ($t$ real) is the line through $\Vec{p}$ in the direction $\Vec{v}$.

    If $\Vec{p}$ and $\Vec{q}$ are distinct points, set $\Vec{v} = \Vec{q} - \Vec{p}$; the preceding paragraph describes the line through $\Vec{p}$ and $\Vec{q}$.

    The "symmetric form" for a line is a system of $(n - 1)$ equations in $n$ variables. The coefficients are any linearly independent set of $(n - 1)$ vectors orthogonal to $\Vec{v}$.

  • If $\Vec{p}$ is a point and $\Vec{v}$, $\Vec{w}$ are non-proportional vectors (non-zero in particular), then the set of points $\Vec{p} + s\Vec{v} + t\Vec{w}$ ($s$, $t$ real) is the plane through $\Vec{p}$ parallel to $\Vec{v}$ and $\Vec{w}$.

    If $\Vec{p}$, $\Vec{q}$, and $\Vec{r}$ are non-collinear points, set $\Vec{v} = \Vec{q} - \Vec{p}$ and $\Vec{w} = \Vec{r} - \Vec{p}$; the preceding paragraph describes the plane through $\Vec{p}$, $\Vec{q}$, and $\Vec{r}$.

    The "symmetric form" for a plane is a system of $(n - 2)$ equations in $n$ variables.

  • If $\Vec{p}$ is a point and $\Vec{v}$ is a non-zero vector, then the set of points $\Vec{x}$ satisfying $\Vec{v} \cdot (\Vec{x} - \Vec{p}) = 0$ is the hyperplane through $\Vec{p}$ with normal vector $\Vec{v}$.

Your original post had the "cross product" tag. It turns out the cross product has no generalization (as a binary operation) to arbitrary dimension; there is, however, an $(n - 1)$-ary "cross product" in $\Reals^{n}$.

Andrew D. Hwang
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