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vectors

Dot & cross products, lines and planes in 3D, angles, intersections, distances.

vector algebra · products · lines · planes

Vector algebra

Components. \(\vv{v}=\langle v_1,v_2,v_3\rangle=v_1\vv{i}+v_2\vv{j}+v_3\vv{k}\); magnitude \(|\vv{v}|=\sqrt{v_1^2+v_2^2+v_3^2}\); unit vector \(\uvec{v}=\vv{v}/|\vv{v}|\).

Position vectors. \(\overrightarrow{OA}=\vv{a}\), \(\overrightarrow{AB}=\vv{b}-\vv{a}\). Parallel iff \(\vv{u}=\lambda\vv{v}\).

Products

Dot product. \(\vv{u}\cdot\vv{v}=u_1v_1+u_2v_2+u_3v_3=|\vv{u}||\vv{v}|\cos\theta\); perpendicular iff \(\vv{u}\cdot\vv{v}=0\).

Projection. \(\operatorname{proj}_{\vv{v}}\vv{u}=\dfrac{\vv{u}\cdot\vv{v}}{|\vv{v}|^2}\,\vv{v}\).

Cross product. \[\vv{u}\times\vv{v}=\begin{vmatrix}\vv{i}&\vv{j}&\vv{k}\\u_1&u_2&u_3\\v_1&v_2&v_3\end{vmatrix},\qquad |\vv{u}\times\vv{v}|=|\vv{u}||\vv{v}|\sin\theta,\] normal to both; parallel iff \(\vv{u}\times\vv{v}=\vv{0}\).

Areas & volume. Parallelogram area \(|\vv{u}\times\vv{v}|\); triangle \(\frac12|\vv{u}\times\vv{v}|\). Scalar triple product: parallelepiped volume \(|\vv{u}\cdot(\vv{v}\times\vv{w})|\).

Lines

Forms. Vector \(\vv{r}=\vv{a}+\lambda\vv{b}\); parametric \(x=x_0+\lambda l\), \(y=y_0+\lambda m\), \(z=z_0+\lambda n\); Cartesian \(\dfrac{x-x_0}{l}=\dfrac{y-y_0}{m}=\dfrac{z-z_0}{n}\) where denominators are nonzero.

Angle between lines. \(\cos\theta=\dfrac{|\vv{b}_1\cdot\vv{b}_2|}{|\vv{b}_1||\vv{b}_2|}\).

Intersection. Solve \(\vv{a}+\lambda\vv{b}=\vv{c}+\mu\vv{d}\). Parallel/coincident/skew classified by whether directions are proportional and whether a solution exists.

Distances. Point to line: \(d=\dfrac{|(\vv{p}-\vv{a})\times\vv{b}|}{|\vv{b}|}\). Between skew lines: \(d=\dfrac{|(\vv{c}-\vv{a})\cdot(\vv{b}\times\vv{d})|}{|\vv{b}\times\vv{d}|}\).

Planes

Forms. Vector \(\vv{r}=\vv{a}+\lambda\vv{b}+\mu\vv{c}\) with \(\vv{b},\vv{c}\) nonparallel; normal form \(\vv{r}\cdot\vv{n}=\vv{a}\cdot\vv{n}=d\), where \(\vv{n}=\vv{b}\times\vv{c}\); Cartesian \(ax+by+cz=d\), normal \(\vv{n}=\langle a,b,c\rangle\).

Angles. Between planes: \(\cos\theta=\dfrac{|\vv{n}_1\cdot\vv{n}_2|}{|\vv{n}_1||\vv{n}_2|}\). Line–plane: \(\sin\theta=\dfrac{|\vv{b}\cdot\vv{n}|}{|\vv{b}||\vv{n}|}\).

Intersections. Line–plane: substitute the line into the plane equation. Plane–plane: the line of intersection has direction \(\vv{n}_1\times\vv{n}_2\). Three planes: solve the \(3\times3\) system — unique point, line, plane, or no solution, distinguished by ranks.

Distance. Point to plane \(\vv{r}\cdot\vv{n}=D\): \(d=\dfrac{|\vv{p}\cdot\vv{n}-D|}{|\vv{n}|}\).