Mathematics

Vectors, lines and planes in ℝ³

Vector bases, operations with 3D vectors, equations of the line, dot product, cross product and scalar triple product, area and volume calculation.

The set of vectors u⃗1,u⃗2,…,u⃗n\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n is said to form a vector base, of dimension n, denoted by B={u⃗1,u⃗2,…,u⃗n}B = \{\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n\} if it verifies:
  • The vectors u⃗1,u⃗2,…,u⃗n\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n are linearly independent.
  • Any other vector v⃗\vec{v} can be expressed as a linear combination of u⃗1,u⃗2,…,u⃗n\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n, that is:
v⃗=λ1u⃗1+λ2u⃗2+⋯+λnu⃗n\vec{v} = \lambda_1\vec{u}_1 + \lambda_2\vec{u}_2 + \dots + \lambda_n\vec{u}_n
Canonical base R3\mathbb{R}^3: is the vector base of dimension 3 formed by the unit vectors:
B={(1,0,0),(0,1,0),(0,0,1)}B = \{(1,0,0), (0,1,0), (0,0,1)\}
Orthogonal base: A base where its vectors are pairwise perpendicular
Orthonormal base: Orthogonal base of unit vectors.
Given 3 vectors, u,v,wu, v, w, they are linearly independent if the determinant formed by the 3 vectors is NON-ZERO
∣uxuyuzvxvyvzwxwywz∣≠0\begin{vmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix} \neq 0
Magnitude of a vector:
∣u⃗∣=ux2+uy2+uz2|\vec{u}| = \sqrt{u_x^2 + u_y^2 + u_z^2}
Unit vector: is a vector of magnitude 1
Normalization of a vector u⃗\vec{u}: is to determine another unit vector, n⃗u\vec{n}_u in the same direction as u⃗\vec{u}
n⃗u=(ux∣u⃗∣,uy∣u⃗∣,uz∣u⃗∣)\vec{n}_u = \left( \frac{u_x}{|\vec{u}|}, \frac{u_y}{|\vec{u}|}, \frac{u_z}{|\vec{u}|} \right)
Midpoint of a line segment:
Midpoint of a line segment
M=(ax+bx2,ay+by2,az+bz2)M = \left( \frac{a_x + b_x}{2}, \frac{a_y + b_y}{2}, \frac{a_z + b_z}{2} \right)
3 collinear points: satisfy
AB⃗=k⋅AC⃗\vec{AB} = k \cdot \vec{AC}
Collinear points
In R3\mathbb{R}^3 they are obtained from a direction vector u⃗=(ux,uy,uz)\vec{u}=(u_x, u_y, u_z) and a point P=(x0,y0,z0)P=(x_0, y_0, z_0)
Vector equation:
(x,y,z)=(x0,y0,z0)+λ(ux,uy,uz)(x,y,z) = (x_0,y_0,z_0) + \lambda(u_x,u_y,u_z)
Parametric equations:
{x=x0+λuxy=y0+λuyz=z0+λuz\begin{cases} x = x_0 + \lambda u_x \\ y = y_0 + \lambda u_y \\ z = z_0 + \lambda u_z \end{cases}
Symmetric equations:
x−x0ux=y−y0uy=z−z0uz\frac{x - x_0}{u_x} = \frac{y - y_0}{u_y} = \frac{z - z_0}{u_z}
Implicit equations: They can be obtained by cross-multiplying two of the equalities of the symmetric equations
{Ax+By+Cz=DA′x+B′y+C′z=D′\begin{cases} Ax + By + Cz = D \\ A'x + B'y + C'z = D' \end{cases}
u⃗⋅v⃗=uxvx+uyvy+uzvz\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y + u_z v_z
or also:
u⃗⋅v⃗=∣u⃗∣⋅∣v⃗∣⋅cos⁡α\vec{u} \cdot \vec{v} = |\vec{u}| \cdot |\vec{v}| \cdot \cos \alpha
Projection of u⃗\vec{u} onto v⃗\vec{v}
Projection of u onto v
Proyv⃗u⃗=u⃗⋅v⃗∣v⃗∣2v⃗\text{Proy}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2} \vec{v}
Orthogonal (or perpendicular) vectors:
u⃗⋅v⃗=0\vec{u} \cdot \vec{v} = 0
u⃗×v⃗=∣ijkuxuyuzvxvyvz∣∣u⃗×v⃗∣=∣u⃗∣⋅∣v⃗∣⋅sin⁡α\begin{aligned} \vec{u} \times \vec{v} &= \begin{vmatrix} i & j & k \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix} \\ \\ |\vec{u} \times \vec{v}| &= |\vec{u}| \cdot |\vec{v}| \cdot \sin \alpha \end{aligned}
The direction of u⃗×v⃗\vec{u} \times \vec{v} is perpendicular to both u⃗\vec{u} and v⃗\vec{v}, and the sense is given by the right-hand rule
Right-hand rule
Area of the parallelogram
Area of the parallelogram
A=∣u⃗×v⃗∣A = |\vec{u} \times \vec{v}|
Area of a triangle
Area of a triangle
A=12∣u⃗×v⃗∣A = \frac{1}{2} |\vec{u} \times \vec{v}|
[u⃗,v⃗,w⃗]=∣uxuyuzvxvyvzwxwywz∣[\vec{u}, \vec{v}, \vec{w}] = \begin{vmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix}
Volume of the parallelepiped
Volume of the parallelepiped
V=∣[u⃗,v⃗,w⃗]∣V = |[\vec{u}, \vec{v}, \vec{w}]|
Volume of the tetrahedron
Volume of the tetrahedron
V=16∣[u⃗,v⃗,w⃗]∣V = \frac{1}{6} |[\vec{u}, \vec{v}, \vec{w}]|
Vector equation: with two direction vectors and a point:
u⃗=(ux,uy,uz),v⃗=(vx,vy,vz),P=(x0,y0,z0)\vec{u} = (u_x, u_y, u_z), \vec{v} = (v_x, v_y, v_z), P = (x_0, y_0, z_0)
(x,y,z)=(x0,y0,z0)+λ(ux,uy,uz)+μ(vx,vy,vz)(x,y,z) = (x_0,y_0,z_0) + \lambda(u_x, u_y, u_z) + \mu(v_x, v_y, v_z)
Parametric equations
{x=x0+λux+μvxy=y0+λuy+μvyz=z0+λuz+μvz\begin{cases} x = x_0 + \lambda u_x + \mu v_x \\ y = y_0 + \lambda u_y + \mu v_y \\ z = z_0 + \lambda u_z + \mu v_z \end{cases}
Implicit equation: with a point and 2 direction vectors
∣x−x0y−y0z−z0uxuyuzvxvyvz∣=Ax+By+Cz+D=0\begin{vmatrix} x - x_0 & y - y_0 & z - z_0 \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix} = Ax + By + Cz + D = 0
Where n⃗=(A,B,C)\vec{n} = (A,B,C) is a normal vector to the plane.
Implicit equation: with a point P=(x0,y0,z0)P=(x_0, y_0, z_0) and a normal vector n⃗=(A,B,C)\vec{n} = (A,B,C)
A(x−x0)+B(y−y0)+C(z−z0)=0A(x - x_0) + B(y - y_0) + C(z - z_0) = 0
Intersecting:
λ(Ax+By+Cz+D)+μ(A′x+B′y+C′z+D′)=0\lambda(Ax + By + Cz + D) + \mu(A'x + B'y + C'z + D') = 0
The axis is the line:
{Ax+By+Cz+D=0A′x+B′y+C′z+D′=0\begin{cases} Ax + By + Cz + D = 0 \\ A'x + B'y + C'z + D' = 0 \end{cases}
Intersecting pencil of planes
Parallel:
Ax+By+Cz+λ=0Ax + By + Cz + \lambda = 0
Parallel pencil of planes