# Lines and Planes in Space

Mathematics I · Matrices and Vectors · https://tryals.app/learn/mathematics-i/lines-and-planes-in-space

Vectors let us describe **lines** and **planes** in any dimension by starting from a fixed point and moving along directions. A **line** through a point with position vector $r_0$ and direction vector $d$ is defined by:

$r = r_0 + t d, \quad t \in \mathbb{R}$

The parameter $t$ slides you along the line, while $d$ sets the heading. Unlike slope-intercept form, this **parametric** form works identically in 2D, 3D, and beyond.

A **plane** is captured by a **normal vector** $n$ that is perpendicular to the plane. A point $r$ lies in the plane through $r_0$ exactly when the displacement is orthogonal to $n$:

$n \cdot (r - r_0) = 0$

Expanded in coordinates, this yields $ax + by + cz = d$, where $(a, b, c)$ is the normal vector.

## Vector Relations

Translating geometry into vector algebra makes spatial tests routine. The relations between planes depend directly on their normal vectors:

| Relationship | Condition on Normals | Vector Test |
| --- | --- | --- |
| **Parallel** | $n_1$ parallel to $n_2$ | $n_1 = k n_2$ |
| **Perpendicular** | $n_1$ orthogonal to $n_2$ | $n_1 \cdot n_2 = 0$ |
| **Angle** | Angle between planes | $\cos \theta = \frac{n_1 \cdot n_2}{|n_1| |n_2|}$ |

> **Common Pitfall:** Never confuse a plane's **normal vector** with a direction lying **in** the plane. The normal is **perpendicular** to the surface and provides the coefficients $(a, b, c)$. Using an in-plane direction instead of the normal completely inverts the geometry.

## Practice questions

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

### 1. Find the value of k for which the planes x + 2y + kz = 1 and 3x - y + 2z = 4 are perpendicular.

**Answer:** -0.5 (within ±0.01)

**Why:** The normals dot to 3 - 2 + 2k = 1 + 2k, and setting that to zero gives k = -0.5. The whole question reduces to a dot product on the coefficient vectors, the payoff of representing planes by their normals.

Page: https://tryals.app/practice/mathematics-i/lines-and-planes-in-space/find-the-value-of-k-for-which-the-planes-x-2y-kz-1-and-3x-y

### 2. The plane 2x - 3y + z = 7 has normal vector (a, b, c). What is b?

**Answer:** -3

**Why:** The normal is (2, -3, 1), so b = -3. The constant 7 plays no part in the direction, it only shifts the plane along that normal. Dropping the minus sign would give a vector pointing into a different half of space.

Page: https://tryals.app/practice/mathematics-i/lines-and-planes-in-space/the-plane-2x-3y-z-7-has-normal-vector-a-b-c-what-is-b

### 3. Select every point that lies on the line given by r = (1, 0, 2) + t(2, 1, -1).

A. (5, 2, 0)
B. (3, 1, 1)
C. (-1, -1, 3)
D. (0, 0, 0)
E. (1, 1, 2)

**Answer:** A. (5, 2, 0); B. (3, 1, 1); C. (-1, -1, 3)

**Why:** t = 1 gives (3, 1, 1), t = -1 gives (-1, -1, 3) and t = 2 gives (5, 2, 0). For (1, 1, 2) the first coordinate forces t = 0, which then demands y = 0, not 1. For the origin the first coordinate needs t = -0.5, which gives y = -0.5. One consistent t must serve every coordinate at once.

Page: https://tryals.app/practice/mathematics-i/lines-and-planes-in-space/select-every-point-that-lies-on-the-line-given-by-r-1-0-2

### 4. A plane has the form 2x + y - z = d and passes through the point (1, 2, 3). Find d.

**Answer:** 1

**Why:** 2(1) + (2) - (3) = 1, so d = 1. The coefficients fix the plane's orientation via the normal (2, 1, -1), and d selects which of the parallel family actually passes through the given point.

Page: https://tryals.app/practice/mathematics-i/lines-and-planes-in-space/a-plane-has-the-form-2x-y-z-d-and-passes-through-the-point-1

### 5. Complete the interpretation of a plane's coefficients.

**Answer:** In the plane equation ax + by + cz = d, the vector (a, b, c) is **the normal vector, perpendicular to the plane**.

**Why:** The coefficients ARE the normal, perpendicular to the plane. The equation is just the dot product condition n . (r - r0) = 0 written out. Mistaking the normal for a direction inside the plane inverts every parallel and perpendicular test.

Page: https://tryals.app/practice/mathematics-i/lines-and-planes-in-space/complete-the-interpretation-of-a-planes-coefficients

### 6. The rank of a matrix can be defined by its rows or by its columns, and the two numbers are always equal, even for a 3 × 700 matrix. Why is that surprising?

A. Because rank is defined only for square matrices, making the comparison ill-posed
B. It is not surprising, the equality follows immediately from the definition
C. Because rows and columns are interchangeable by transposition, so the equality is trivial
D. Because the two counts are taken over completely different spaces

**Answer:** D. Because the two counts are taken over completely different spaces

**Why:** Three vectors in ℝ⁷⁰⁰ and seven hundred in ℝ³, and the number of independent ones matches every time. It is a genuine theorem rather than a restatement, and it is what makes rank a property of the transformation rather than of how you chose to look at the array.

Page: https://tryals.app/practice/mathematics-i/lines-and-planes-in-space/the-rank-of-a-matrix-can-be-defined-by-its-rows-or-by-its-columns

### 7. The same matrix equation Ax = b describes a system of equations, a transformation applied to a vector, and a combination of A's columns. Why is having three readings of one equation useful rather than merely a curiosity?

A. Because the three readings apply to different kinds of matrix, and one must choose the correct one for the problem at hand.
B. Because the readings give different answers, and comparing them checks the arithmetic.
C. Because only the row reading is mathematically rigorous, the others being informal aids to intuition.
D. Because each reading makes a different question easy.

**Answer:** D. Because each reading makes a different question easy.

**Why:** The three readings are the same mathematics from different vantage points, and each makes a different question easy. Column reading answers existence at once, a solution exists precisely when b lies in the span of the columns. Transformation reading answers uniqueness, since anything in the kernel can be added to a solution. Fluency in switching is most of practical linear algebra.

Page: https://tryals.app/practice/mathematics-i/lines-and-planes-in-space/the-same-matrix-equation-ax-b-describes-a-system-of-equations-a

### 8. Sort each description by the object it defines.

**Answer:**

- Describes a line: r = r0 + t d, (x, y, z) = (1, 2, 3) + t(0, 1, 1)
- Describes a plane: n . (r - r0) = 0, 2x + 3y - z = 5, The points whose displacement from a fixed point is orthogonal to a fixed vector

**Why:** One parameter t gives one degree of freedom, hence a line. A single dot-product condition removes one degree of freedom from space, leaving two, a plane. The last item is just the geometric description of n . (r - r0) = 0 in words.

Page: https://tryals.app/practice/mathematics-i/lines-and-planes-in-space/sort-each-description-by-the-object-it-defines