Lesson 3-2: Spatial Coordinates & Line-Plane Relations

Learning Objectives

Compute 3D distances and midpoints from coordinate definitions.
Write line equations in parametric/symmetric forms using direction vectors.
Construct plane equations from point-normal or three-points forms.

Determine intersections, parallelism, and perpendicular relations.
Find angles between lines and planes; compute point-plane distances.

Core Concepts

Distances & Midpoints

d(P_1,P_2)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

M=\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2},\tfrac{z_1+z_2}{2}\right)

Line Equations

\text{Parametric: }\; \vec{r}=\vec{r}_0 + t\,\vec{v}

\text{Symmetric: }\; \tfrac{x-x_0}{v_x}=\tfrac{y-y_0}{v_y}=\tfrac{z-z_0}{v_z}

Plane Equations

\text{Point-normal: }\; \vec{n}\cdot(\vec{r}-\vec{r}_0)=0

\text{General: }\; Ax+By+Cz+D=0

Angles & Distances

\cos\theta = \tfrac{\vec{u}\cdot\vec{v}}{\|\vec{u}\|\,\|\vec{v}\|}

d(P,\Pi)=\tfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}

Worked Examples

Example 1: Distance & Midpoint

For A(2,1,3), B(4,-1,5), compute d(A,B) and midpoint.

d=\sqrt{(4-2)^2+(-1-1)^2+(5-3)^2}=\sqrt{12}=2\sqrt{3}

M=\left(\tfrac{2+4}{2},\tfrac{1+(-1)}{2},\tfrac{3+5}{2}\right)=\left(3,0,4\right)

Example 2: Line with Plane

Plane $2x-y+3z-4=0$ , line through P(1,2,-1) with direction $\vec{v}=(1,2,-1)$ . Find intersection.

x=1+t,\; y=2+2t,\; z=-1-t

2(1+t)-(2+2t)+3(-1-t)-4=0 \Rightarrow -3t-7=0 \Rightarrow t=-\tfrac{7}{3}

X=\left(1-\tfrac{7}{3},\;2+2(-\tfrac{7}{3}),\;-1-(-\tfrac{7}{3})\right)=\left(-\tfrac{4}{3},-\tfrac{8}{3},\tfrac{4}{3}\right)

Example 3: Parallel / Skew / Intersect

Determine the relation between $l_1: \vec{r}=\vec{a}+t\vec{u}$ and $l_2: \vec{r}=\vec{b}+s\vec{v}$ .

Strategy: check if directions are proportional; if not, set equations equal and test for a common solution.

• Parallel: $\vec{u} \parallel \vec{v}$ , no intersection unless collinear.
• Intersect: exists (t,s) s.t. $\vec{a}+t\vec{u}=\vec{b}+s\vec{v}$ .
• Skew: not parallel and no intersection.

Example 4: Angle Between Line and Plane

For line direction $\vec{v}=(1,2,-1)$ and plane normal $\vec{n}=(2,-1,3)$ , find angle $\varphi$ .

\sin\varphi = \frac{|\vec{v}\cdot \vec{n}|}{\|\vec{v}\|\,\|\vec{n}\|} = \frac{3}{\sqrt{6}\,\sqrt{14}} = \frac{3}{2\sqrt{21}}

Thus $\varphi \approx 23.4^\circ$ .

Example 5: Distance from Point to Plane

Distance from P( $x_0,y_0,z_0$ ) to $Ax+By+Cz+D=0$ .

d=\tfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}

Practice Problems

Problem 1

Find the equation of the plane through A(1,0,0), B(0,2,0), C(0,0,3).

Show Solution

Use intercept form: $\tfrac{x}{1}+\tfrac{y}{2}+\tfrac{z}{3}=1$ ⇒ $6x+3y+2z=6$ .

Problem 2

Find intersection point of line $x=2+t,\;y=1-2t,\;z=3+t$ with plane $x+y+z=6$ .

Show Solution

Substitute: $(2+t)+(1-2t)+(3+t)=6$ ⇒ $6=6$ ⇒ identity; the line lies in the plane.

Problem 3

Angle between planes $2x-y+3z-4=0$ and $x+2y-z+1=0$ .

Show Solution

Angle equals angle between normals: $\cos\theta=\tfrac{|(2,-1,3)\cdot(1,2,-1)|}{\sqrt{2^2+(-1)^2+3^2}\,\sqrt{1^2+2^2+(-1)^2}}=\tfrac{|2-2-3|}{\sqrt{14}\,\sqrt{6}}=\tfrac{3}{\sqrt{84}}$ .

Problem 4

Distance from P(1,2,-1) to plane $2x-y+3z-4=0$ .

Show Solution

$d=\tfrac{|2(1)-1(2)+3(-1)-4|}{\sqrt{2^2+(-1)^2+3^2}}=\tfrac{|2-2-3-4|}{\sqrt{14}}=\tfrac{7}{\sqrt{14}}$ .

Problem 5

Find symmetric equation of a line through (1,2,3) parallel to vector (2,-1,4).

Show Solution

$\tfrac{x-1}{2}=\tfrac{y-2}{-1}=\tfrac{z-3}{4}$ .

Key Takeaways

Use vectors to unify line/plane representations and computations.
Distances and angles reduce to dot products and norms.
Parallel: normals or directions are proportional; perpendicular: dot product 0.
Check feasibility in intersection problems; beware of skew lines.

Extended Topics

Angle Between Two Planes

Angle equals the angle between the normals.

\cos\theta=\tfrac{|\vec{n}_1\cdot\vec{n}_2|}{\|\vec{n}_1\|\,\|\vec{n}_2\|}

Distance Between Skew Lines

Use cross product of directions to find shortest distance.

d=\tfrac{|(\vec{b}-\vec{a})\cdot(\vec{u}\times \vec{v})|}{\|\vec{u}\times \vec{v}\|}

Vector Projections & Components

Projection of u onto v

\operatorname{proj}_{\vec{v}}(\vec{u}) = \dfrac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|^2}\, \vec{v}

Decompose $\vec{u}=\vec{u}_{\parallel}+\vec{u}_{\perp}$ relative to $\vec{v}$ .

Work and Components

Work $W=\vec{F}\cdot\vec{s}=\|\vec{F}\|\,\|\vec{s}\|\cos\theta$ ; projection clarifies effective force along displacement.

Coordinate Transformations

Rotation about z-axis

\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}\cos\phi & -\sin\phi & 0 \ \ \sin\phi & \cos\phi & 0 \ \ 0 & 0 & 1\end{pmatrix} \begin{pmatrix}x\\y\\z\end{pmatrix}

To Cylindrical

r=\sqrt{x^2+y^2},\; \theta=\operatorname{atan2}(y,x),\; z=z

Practice Banks

Bank 1: Line Forms

Convert parametric ↔ symmetric forms.
Identify direction vector and a point.
Check equivalence conditions.

Bank 2: Line–Plane Intersection

Substitute parametric line into plane.
Solve for parameter; classify relation.
Report intersection or parallelism.

Bank 3: Distance & Projection

Compute point–plane distance.
Project vectors and interpret.
Explain geometric meaning.

Bank 4: Angles

Angles line–line and plane–plane.
Angles line–plane via sine formula.
Check units and rounding.

Proof Sketches & FAQ

Why Distance Formula Works

Use Pythagorean theorem in 3D by successive orthogonal projections.

FAQ

Skew lines vs parallel? Non-coplanar with no intersection vs same direction.
Parametric speed? Magnitude of direction vector scales parameterization.
Multiple forms of plane? Equivalent given same normal and passing points.

Extended Example Sets

Example 6: Intersection of Two Planes

Find the intersection line of $Pi_1$ and $Pi_2$ , then angle with a given line.

Example 7: Distance Between Skew Lines

Use cross product of directions to compute shortest distance.

Example 8: Projection in Physics

Resolve force along an inclined plane using projections.

Example 9: Rotation Matrix Application

Rotate coordinates about z-axis and verify distances preserved.

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