Generalize Formulas to $\mathbb{R}^n$ from formulas from vectors in 3D

Question

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 $\mathbb{R^{3}}$.

Answer 1

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}$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 $\Reals^{n}$.