Applications of Vector Algebra

Applications of Vector Algebra — Formula Sheet

Every key identity and formula for the chapter, grouped by topic.

\([\vec{a},\vec{b},\vec{c}]=(\vec{a}\times\vec{b})\cdot\vec{c}=\vec{a}\cdot(\vec{b}\times\vec{c})\)
\([\vec{a},\vec{b},\vec{c}]=\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}\)
Volume of parallelepiped \(=\big|[\vec{a},\vec{b},\vec{c}]\big|\)
Cyclic: \([\vec{a},\vec{b},\vec{c}]=[\vec{b},\vec{c},\vec{a}]=[\vec{c},\vec{a},\vec{b}]\); one swap changes the sign
Coplanar \(\iff [\vec{a},\vec{b},\vec{c}]=0\)
System of vectors: \([\vec{p},\vec{q},\vec{r}]=\begin{vmatrix}x_{1}&y_{1}&z_{1}\\x_{2}&y_{2}&z_{2}\\x_{3}&y_{3}&z_{3}\end{vmatrix}[\vec{a},\vec{b},\vec{c}]\) when \(\vec{p}=x_{1}\vec{a}+y_{1}\vec{b}+z_{1}\vec{c}\), etc.

\(\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}\)
\((\vec{a}\times\vec{b})\times\vec{c}=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{b}\cdot\vec{c})\vec{a}\)
Jacobi: \(\vec{a}\times(\vec{b}\times\vec{c})+\vec{b}\times(\vec{c}\times\vec{a})+\vec{c}\times(\vec{a}\times\vec{b})=\vec{0}\)
Lagrange: \((\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})=(\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d})-(\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c})\)
\([\vec{a}\times\vec{b},\ \vec{b}\times\vec{c},\ \vec{c}\times\vec{a}]=[\vec{a},\vec{b},\vec{c}]^{2}\)

Point–direction (vector): \(\vec{r}=\vec{a}+t\vec{b}\)
Point–direction (non‑parametric): \((\vec{r}-\vec{a})\times\vec{b}=\vec{0}\)
Cartesian: \(\dfrac{x-x_{1}}{b_{1}}=\dfrac{y-y_{1}}{b_{2}}=\dfrac{z-z_{1}}{b_{3}}\)
Through two points: \(\vec{r}=\vec{a}+t(\vec{b}-\vec{a})\)
Direction cosines: \(l^{2}+m^{2}+n^{2}=1\)
Angle between two lines: \(\cos\theta=\dfrac{|\vec{b}\cdot\vec{d}|}{|\vec{b}|\,|\vec{d}|}\)
Shortest distance (skew): \(d=\dfrac{\big|(\vec{c}-\vec{a})\cdot(\vec{b}\times\vec{d})\big|}{|\vec{b}\times\vec{d}|}\)
Shortest distance (parallel): \(d=\dfrac{\big|(\vec{c}-\vec{a})\times\vec{b}\big|}{|\vec{b}|}\)
Coplanarity of two lines: \((\vec{c}-\vec{a})\cdot(\vec{b}\times\vec{d})=0\)

Normal form: \(\vec{r}\cdot\hat{d}=p\) / \(lx+my+nz=p\)
Point–normal: \((\vec{r}-\vec{a})\cdot\vec{n}=0\) / \(a(x-x_{1})+b(y-y_{1})+c(z-z_{1})=0\)
Intercept form: \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\)
Three points: \([\vec{r}-\vec{a},\ \vec{b}-\vec{a},\ \vec{c}-\vec{a}]=0\)
Point and two vectors: \(\vec{r}=\vec{a}+s\vec{b}+t\vec{c}\), i.e. \([\vec{r}-\vec{a},\ \vec{b},\ \vec{c}]=0\)
General: \(ax+by+cz+d=0\), normal \((a,b,c)\)
Plane through intersection: \((\vec{r}\cdot\vec{n}_{1}-d_{1})+\lambda(\vec{r}\cdot\vec{n}_{2}-d_{2})=0\)

Angle between planes: \(\cos\theta=\dfrac{|\vec{n}_{1}\cdot\vec{n}_{2}|}{|\vec{n}_{1}|\,|\vec{n}_{2}|}\)
Angle between line and plane: \(\sin\theta=\dfrac{|\vec{b}\cdot\vec{n}|}{|\vec{b}|\,|\vec{n}|}\)
Point to plane: \(\delta=\dfrac{|\vec{u}\cdot\vec{n}-p|}{|\vec{n}|}=\dfrac{|ax_{1}+by_{1}+cz_{1}-p|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)
Origin to plane: \(\delta=\dfrac{|d|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)
Between parallel planes: \(\delta=\dfrac{|d_{1}-d_{2}|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)
Meeting point of line and plane: \(\vec{u}=\vec{a}+\dfrac{p-(\vec{a}\cdot\vec{n})}{\vec{b}\cdot\vec{n}}\,\vec{b}\), \(\vec{b}\cdot\vec{n}\neq0\)
Image of a point in a plane: \(\vec{v}=\vec{u}+\dfrac{2\,[\,p-(\vec{u}\cdot\vec{n})\,]}{|\vec{n}|^{2}}\,\vec{n}\)