Vectors, Lines, and Planes in Three-Dimensional Space

Vectors in Three-Dimensional Space

Dot Product

The dot product is a fundamental operation in vector algebra, used to determine the angle between two vectors and to project one vector onto another.

• Definition: For two nonzero vectors u and v in two or three dimensions, 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 vectors $u = \langle u_1, u_2, u_3 \rangle$ 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 $u \cdot v = 0$, then u and v are orthogonal (perpendicular).

Projection of a Vector

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

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

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

• Application: Used in decomposing forces, finding shadows, and resolving vectors.

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

• Component Expansion: $\vec{u} \times \vec{v} = (u_2 v_3 - u_3 v_2) \vec{i} - (u_1 v_3 - u_3 v_1) \vec{j} + (u_1 v_2 - u_2 v_1) \vec{k}$

• Application: Used to find area of parallelograms and triangles, and to determine perpendicular directions.

Area of a Triangle Using Cross Product

The area of a triangle defined by three points $P$, $Q$, and $R$ in space can be found using the cross product.

• Formula: $\text{Area} = \frac{1}{2} \| \vec{PQ} \times \vec{PR} \|$

• Example: For $P=(1,0,1)$, $Q=(2,2,4)$, $R=(2,1,2)$, compute $\vec{PQ}$ and $\vec{PR}$, then their cross product, and take half its magnitude.

Equations of Lines in Space

Vector and Parametric Equations of a Line

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 the position vector of a fixed point, $\vec{v}$ is a direction vector, and $t$ is a real parameter.

• Parametric Equations: $x = x_0 + a t$ $y = y_0 + b t$ $z = z_0 + c t$ for $-\infty < t < \infty$.

• Example: The line through $P=(2,3,4)$ and $Q=(1,1,1)$ has direction vector $\vec{v} = (-1,-2,-3)$ and parametric equations $x=2-t$, $y=3-2t$, $z=4-3t$.

Distance Between a Point and a Line

The shortest distance from a point to a line in space can be calculated using the cross product.

• Formula: $d = \frac{\| \vec{v} \times \vec{PQ} \|}{\| \vec{v} \|}$ where $\vec{v}$ is the direction vector of the line, $P$ is a point on the line, and $Q$ is the external point.

• Application: Useful in geometry and physics for finding minimum distances.

Equations of Planes in Space

Standard and General Form of a Plane

A plane in three-dimensional space is defined 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 $(a, b, c)$ is the normal vector.

• General Form: $a x + b y + c z = d$ where $d = a x_0 + b y_0 + c z_0$.

• Example: The $xy$-plane is given by $z=0$ with normal vector $(0,0,1)$.

Finding the Equation of a Plane

To find the equation of a plane passing through three points, use the cross product to determine the normal vector.

• Steps:

  1. Find two vectors in the plane (e.g., $\vec{OP}$ and $\vec{OQ}$).

  2. Compute their cross product to get the normal vector $\vec{n}$.

  3. Use the standard form with one of the points.

• Example: For points $P=(1,-2,1)$, $Q=(-1,0,1)$, and $O=(0,0,0)$, the normal vector is $(-2,-2,-2)$, so the plane equation is $x+y+z=0$.

Parallel and Orthogonal Planes

Planes can be classified based on their normal vectors.

• Parallel Planes: Two planes are parallel if their normal vectors are scalar multiples of each other.

• Orthogonal Planes: Two planes are orthogonal if the dot product of their normal vectors is zero.

Intersections of Planes

Intersection of Two Planes

The intersection of two planes in space is typically a line, unless the planes are parallel or coincident.

• Method: Solve the system of equations given by the two plane equations.

• Example: The intersection of $x+2y+3z=0$ and $x+y+z=1$ is the line $(2,-1,0)+t(1,-2,1)$.

Additional Geometric Context

Visualizing Cross Product and Area

The cross product can be visualized as the area of a parallelogram formed by two vectors, and half that area gives the area of a triangle.

• Application: Used in physics and engineering to compute torque, angular momentum, and areas.