# Lines, Planes, and Geometric Transformations

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/lines-planes-and-geometric-transformations

## Lines and Planes in Space

Geometry in 3D space relies on fundamental equations for lines and planes. A **Line in 3D** is defined by a starting point $P_0$ and direction vector $\vec{d}$ using parameter $t$:

$$\vec{r}(t) = P_0 + t\,\vec{d}$$

A **Plane in 3D** uses a point $P_0$ and a normal vector $\vec{n}=(a,b,c)$ where $ax+by+cz = d$.

**Common Pitfall:** A plane's normal vector is perpendicular to every vector **in** the plane. Its components come from the equation coefficients ($a,b,c$), never from points on the plane itself.

## Distances and Transformations

Measure spatial relationships using core formulas. The **Distance** from point $(x_0,y_0,z_0)$ to a plane is:

$$D = \frac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}}$$

| Concept | Formula / Rule | Notes |
| --- | --- | --- |
| **Plane Angle** | $\cos\theta = \frac{|\vec{n}_1\cdot\vec{n}_2|}{\|\vec{n}_1\|\|\vec{n}_2\|}$ | Angle between normals |
| **Parallel Planes** | $\vec{n}_1 \times \vec{n}_2 = \vec{0}$ | Cross product is zero |
| **Line-Plane** | Substitute $\vec{r}(t)$ | Solve for $t$ |

**Geometric transformations** compose as matrix products. Using **homogeneous coordinates**, rotation and translation combine into a single matrix multiplication.

## Practice questions

6 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Which are possible relationships between two distinct planes in ℝ³?

A. Identical
B. Parallel (no intersection)
C. Intersect in a point
D. Intersect in a line

**Answer:** A. Identical; B. Parallel (no intersection); D. Intersect in a line

**Why:** Two distinct planes in ℝ³ are either parallel or intersect along a line, never at a single point.

Page: https://tryals.app/practice/physics-i/lines-planes-and-geometric-transformations/which-are-possible-relationships-between-two-distinct-planes-in

### 2. Order the steps to find the equation of a plane through 3 points.

**Answer:**

1. Compute two vectors in the plane: $v_{1} = P_{2}-P_{1}, v_{2} = P_{3}-P_{1}$
2. Find the normal: $n = v_{1} \times v_{2}$
3. Use point-normal form: n $\cdot (r$ − $P_{1}$) = 0
4. Expand to ax + by + cz = d

**Why:** The cross product of two in-plane vectors gives the normal direction.

Page: https://tryals.app/practice/physics-i/lines-planes-and-geometric-transformations/order-the-steps-to-find-the-equation-of-a-plane-through-3-points

### 3. A plane has normal vector $\vec{n} = (1, 2, 2)$. A drone flies along direction $\vec{d} = (2, 1, -2)$. Will its path ever pierce the plane?

A. Yes, every line eventually crosses every plane
B. Not necessarily: the flight is parallel to the plane
C. No, a line meets a plane unless it lies inside it
D. Yes, because $\vec{d}$ and $\vec{n}$ are not equal

**Answer:** B. Not necessarily: the flight is parallel to the plane

**Why:** $\vec{d}\cdot\vec{n} = 0$ means the flight direction is perpendicular to the normal, hence parallel to the plane: the drone skims alongside forever (or lies inside it). One dot product settles an entire geometric question, the normal vector is the plane’s spokesperson.

Page: https://tryals.app/practice/physics-i/lines-planes-and-geometric-transformations/a-plane-has-normal-vector-n-1-2-2-a-drone-flies-along

### 4. Match each geometric question about lines and planes to the algebraic test that answers it.

**Answer:**

- Are two planes parallel? → Are their normal vectors proportional?
- Is a line parallel to a plane? → Is direction · normal = 0?
- Are two lines perpendicular? → Is direction · direction = 0?
- Distance from a point to a plane? → Project the offset onto the unit normal

**Why:** Analytic geometry is a dictionary: parallel ↔ proportional vectors, perpendicular ↔ zero dot product, distance ↔ projection. Fluency means hearing the geometric question and reaching for its algebraic translation without pausing.

Page: https://tryals.app/practice/physics-i/lines-planes-and-geometric-transformations/match-each-geometric-question-about-lines-and-planes-to-the-algebraic

### 5. Two lines in $\mathbb{R}^3$ can be neither parallel nor intersecting.

**Answer:** True

**Why:** True. **Skew** lines run in different directions and never meet, impossible in the plane, routine in space, as with two edges on opposite faces of a cube. It is a good check on geometric intuition, since the planar case makes 'not parallel' feel like it should force a crossing.

Page: https://tryals.app/practice/physics-i/lines-planes-and-geometric-transformations/two-lines-in-can-be-neither-parallel-nor-intersecting

### 6. A line is parallel to a plane when its direction dotted with the plane’s normal is zero. Why does perpendicularity to the NORMAL mean parallelism to the plane?

A. Because the normal is parallel to the plane by definition
B. Because a plane can be described by many different normals, so no single test applies
C. Because the dot product is zero only for parallel vectors
D. Because the normal points out, so perpendicular means within

**Answer:** D. Because the normal points out, so perpendicular means within

**Why:** The normal encodes the whole plane in one vector. This is what makes plane geometry computable, every question about the surface becomes a dot product with its normal.

Page: https://tryals.app/practice/physics-i/lines-planes-and-geometric-transformations/a-line-is-parallel-to-a-plane-when-its-direction-dotted-with-the