Unit 8: Applications of Vectors in Spatial Geometry

The vector formalism condenses three-dimensional analytical geometry into coordinate-free, coordinate-independent equations.

1. Vector Equations of a Straight Line

1. Line through a Point with a Given Direction:

Passing through $\vec{a}$ parallel to $\vec{b}$:

In non-parametric form (since $\vec{r} - \vec{a} \parallel \vec{b}$):

2. Line Passing Through Two Points:

Passing through $\vec{a}$ and $\vec{b}$:

2. Vector Equations of a Plane

1. Point-Normal Form:

Passing through $\vec{a}$ with normal vector $\vec{n}$:

2. Hesse Normal Form:

Using unit normal $\hat{n}$ pointing away from the origin, $p$ is the perpendicular distance from origin:

3. Plane Through Three Non-Collinear Points:

Passing through $\vec{a}, \vec{b}, \vec{c}$. The vectors $\vec{r} - \vec{a}$, $\vec{b} - \vec{a}$, and $\vec{c} - \vec{a}$ are coplanar:

Expanding:

3. Intersection of a Vector Line and a Vector Plane

Let the line be $\vec{r} = \vec{a} + t\vec{d}$ and the plane be $\vec{r} \cdot \vec{n} = q$.

Substitute the line parametrization into the plane equation:

If $\vec{d} \cdot \vec{n} \neq 0$ (line is not parallel to the plane):

Substituting this unique parameter $t$ into $\vec{r}(t)$ yields the exact coordinates of the piercing point.

Vector cross and dot products provide concise derivations of all spatial distance formulas.

1. Distance from a Point to a Straight Line

Let $P(\vec{p})$ be a point in space, and let the line be $\mathcal{L}: \vec{r} = \vec{a} + t\vec{d}$.

Consider the vector $\vec{aP} = \vec{p} - \vec{a}$. The area of the parallelogram formed by $\vec{p} - \vec{a}$ and the direction vector $\vec{d}$ is:

On the other hand, $\text{Area} = \text{base} \times \text{height} = |\vec{d}| \cdot D$, where $D$ is the perpendicular distance from $P$ to the line. Equating:

2. Distance from a Point to a Plane

Let $P(\vec{p})$ be a point and let the plane be $\Pi: \vec{r} \cdot \vec{n} = d$.

Let $A(\vec{a})$ be any point on the plane, so that $\vec{a} \cdot \vec{n} = d$. The perpendicular distance $D$ is the absolute value of the scalar projection of $\vec{p} - \vec{a}$ onto the normal vector $\vec{n}$:

3. Angle Between a Line and a Plane

Let the line have direction $\vec{d}$ and the plane have normal $\vec{n}$. The angle $\theta$ between the line and the plane is the complement of the angle between $\vec{d}$ and $\vec{n}$:

1. Definition of the Reciprocal System

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors in $\mathbb{R}^3$, so that their scalar triple product is non-zero:

The reciprocal triad (or dual basis) of vectors, denoted by $\vec{a}', \vec{b}', \vec{c}'$, is defined as:

2. Fundamental Orthogonality and Normalization Relations

Theorem: For the direct triad $(\vec{a}_1, \vec{a}_2, \vec{a}_3) = (\vec{a}, \vec{b}, \vec{c})$ and the reciprocal triad $(\vec{a}'_1, \vec{a}'_2, \vec{a}'_3) = (\vec{a}', \vec{b}', \vec{c}')$:

Proof:

For $i = j = 1$:

For $i \neq j$, say $\vec{a} \cdot \vec{b}'$:

since any determinant with two identical vectors vanishes identically. $\blacksquare$

3. Reciprocal Volume Theorem

Theorem: The scalar triple product of the reciprocal vectors is the reciprocal of the scalar triple product of the original vectors:

Proof:

By definition:

Using the product of four vectors identity $(\vec{u} \times \vec{v}) \times \vec{w} = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{v} \cdot \vec{w})\vec{u}$ with $\vec{w} = \vec{a} \times \vec{b}$:

Substitute this back:

4. Expansion of an Arbitrary Vector

Any vector $\vec{r} \in \mathbb{R}^3$ can be decomposed immediately in terms of either basis without inverting matrices:

This property makes reciprocal triads indispensable in solid-state physics and crystallography for defining reciprocal lattices.