Vectors, Lines, and Planes in Three-Dimensional Space

Vectors and Their Products

Dot Product

The dot product is a fundamental operation on vectors in two or three dimensions, measuring the extent to which two vectors point in the same direction.

• Definition: For vectors u and v, the dot product is given by: $u \cdot v = \|u\|\|v\| \cos \theta$, where $\theta$ is the angle between u and v, $0 \leq \theta \leq \pi$.

• Component Formula: For $u = \langle u_1, u_2, u_3 \rangle$ and $v = \langle v_1, v_2, v_3 \rangle$, $u \cdot v = u_1 v_1 + u_2 v_2 + u_3 v_3$

• Orthogonality: If $u \cdot v = 0$ and neither vector is zero, then u and v are orthogonal (perpendicular).

Projection of a Vector

The projection of vector u onto vector v is the component of u in the direction of v.

• Formula: $\mathrm{proj}_v u = \left( \frac{u \cdot v}{v \cdot v} \right) v$

• Scalar Component: $\mathrm{scal}_v u = \frac{u \cdot v}{\|v\|}$

Cross Product

The cross product of two vectors in three dimensions produces a vector orthogonal to both.

• Formula: For $\vec{u} = u_1 \vec{i} + u_2 \vec{j} + u_3 \vec{k}$ and $\vec{v} = v_1 \vec{i} + v_2 \vec{j} + v_3 \vec{k}$, $\vec{u} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$

• Geometric Meaning: The magnitude $\|\vec{u} \times \vec{v}\|$ equals the area of the parallelogram spanned by u and v.

Example: Area of a Triangle

Given points $P(1,0,1)$, $Q(2,2,4)$, and $R(2,1,2)$, the area of triangle $PQR$ is:

• Let $\vec{v} = \overrightarrow{PQ}$ and $\vec{w} = \overrightarrow{PR}$.

• Area $= \frac{1}{2} \|\vec{v} \times \vec{w}\|$

Equations of Lines in Space

Vector and Parametric Equations

A line in three-dimensional space can be described using a point and a direction vector.

• Vector Equation: $\vec{r} = \vec{r}_0 + t\vec{v}$, where $\vec{r}_0$ is a position vector to a point on the line, $\vec{v}$ is a direction vector, and $t$ is a real parameter.

• Parametric Equations: $x = x_0 + at$ $y = y_0 + bt$ $z = z_0 + ct$ for $-\infty < t < \infty$.

Example: Line Through Two Points

Given $P(2,3,4)$ and $Q(1,1,1)$, the direction vector is $\overrightarrow{PQ} = (-1,-2,-3)$. The line is:

• Vector form: $\vec{r} = (2,3,4) + t(-1,-2,-3)$

• Parametric: $x = 2-t$, $y = 3-2t$, $z = 4-3t$

Distance from a Point to a Line

The shortest distance from a point $Q$ to a line $\vec{r} = \vec{r}_0 + t\vec{v}$ is:

• $d = \frac{\|\vec{v} \times \overrightarrow{PQ}\|}{\|\vec{v}\|}$, where $P$ is any point on the line.

Equations of Planes in Space

Standard and General Forms

A plane in $\mathbb{R}^3$ is determined by a point and a normal vector.

• Standard Form: $a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$, where $(x_0, y_0, z_0)$ is a point on the plane and $\vec{n} = (a, b, c)$ is a normal vector.

• General Form: $ax + by + cz = d$, where $d = a x_0 + b y_0 + c z_0$.

Examples

• Plane through the origin: $ax + by + cz = 0$

• xy-plane: $z = 0$ (normal vector $\vec{n} = (0,0,1)$)

• Given normal $\vec{n} = (1,1,1)$ and point $Q(5,-3,7)$: $x + y + z = 9$

Finding a Plane from Three Points

Given three points, the normal vector can be found by taking the cross product of two vectors in the plane.

• Let $v = \overrightarrow{OP}$ and $w = \overrightarrow{OQ}$, then $\vec{n} = v \times w$

• Equation: $n_1 x + n_2 y + n_3 z = 0$ (if the plane passes through the origin)

Intersections and Relationships Between Planes

Parallel and Orthogonal Planes

Planes in space can be classified by the relationship between their normal vectors.

• Parallel Planes: Normal vectors are scalar multiples of each other.

• Orthogonal Planes: Normal vectors are orthogonal, i.e., $\vec{n}_1 \cdot \vec{n}_2 = 0$.

Intersection of Two Planes

The intersection of two non-parallel planes is a line. To find the intersection, solve the system of plane equations.

• Example: For $x + 2y + 3z = 0$ and $x + y + z = 1$, solve for $x$, $y$, and $z$ in terms of a parameter to get the line of intersection.

Geometric Interpretation of Vector Products

Area and Orientation

The cross product not only gives a vector orthogonal to two vectors but also encodes the area of the parallelogram they span and the orientation (right-hand rule).