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C. Geometry and Algebra of Vectors · Lines and Planes

Lines and Planes: Unit Review

C3.1 C3.2 C3.3 C4.1 C4.2 C4.3 C4.4 C4.5 C4.6 C4.7About 40 minutes
Learning goals
  • I can describe lines and planes with scalar, vector, parametric and symmetric equations and switch between forms.
  • I can find and classify the intersections of lines and planes, algebraically and geometrically.
  • I can find distances between points, lines and planes.

Words to know

normal
A vector perpendicular to a line (in two-space) or a plane.
skew lines
Non-parallel lines in three-space that do not intersect.
triple scalar product
a · (b × c); zero exactly when the three vectors are coplanar.

Let's review!

In this unit you described lines with scalar (two-space only), vector, parametric and symmetric equations, and planes with scalar, vector and parametric equations. You read normals straight from equations, and used the cross product to build normals and directions. You sorted every arrangement of two lines, a line and a plane, two planes and three planes by how they intersect, solved systems of three equations by elimination, and finished with distances from points to lines and planes, and between skew lines.

This is the last unit of Calculus and Vectors. The ideas continue directly into first-year university linear algebra, and into multivariable calculus, physics, engineering and computer graphics. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Work through one or two sections at a time. Sketch whenever you can: a quick picture of two planes or two lines tells you what kind of answer to expect (a point? a line? nothing?). Always check your final answer: substitute a point into every equation it should satisfy, or check that a direction vector is perpendicular to a normal. For every question you miss, write down why (wrong normal? sign slip? forgot the third equation?) and go back to that lesson. Finish with the 10-question quiz.

Lines1 Systems in 2-space2 Vector/parametric (2-D)3 Connecting the forms4 Lines in 3-spacePlanes5 Planes from equations6 Normals, scalar form7 Vector/parametric formIntersections8 Two planes9 Two lines (skew!)10 Line and plane11 Three planes: pictures12 Three planes: algebraDistances13 Point to line, plane14 Skew lines, line-planeKey tools: normal n = [A, B, C], direction m, cross product n₁ × n₂ or m₁ × m₂,substitution of parametric equations, elimination, and projection onto n.
The four parts of the unit and the lessons in each. The review sections below follow the same order.
Just for fun

Why did the line never make it to the plane's party? It was parallel, so it could travel forever and never arrive. (If it had been lying in the plane, it would have been the life of the party: present at every point.)

Lines and Linear Systems in Two-Space

1

Classify the system 3x − y = 4 and −6x + 2y = −8.

2

Solve 2x + y = 7 and x − 3y = 7.

3

For which k is kx + 4y = 2 parallel to 3x + 6y = 1? Are the lines distinct?

Vector and Parametric Equations of a Line in Two-Space

1

Find a vector equation of the line through A(−2, 5) and B(1, −1).

2

Is (4, −7) on the line in the previous question?

3

Which point on r = [4, 2] + t[2, −6] has t = −1/2?

Connecting Scalar, Vector, Parametric and Symmetric Equations of a Line

1

Find a scalar equation of the line x = 3 − 2t, y = 1 + 4t.

2

Write a vector equation of 4x − y + 8 = 0.

3

Where does 2x + y − 7 = 0 meet the line x = t, y = t + 1?

Vector, Parametric and Symmetric Equations of a Line in Three-Space

1

Write parametric equations of the line through (2, 0, −3) and (5, 2, −1).

2

Write symmetric equations of the line through (1, −4, 0) with direction [2, 0, −3].

3

Where does r = [3, 1, −2] + t[1, 1, 2] cross the xy-plane?

Linear Equations in Three Variables: Planes

1

Find the intercepts of 4x − 2y + z = 8.

2

Describe x + y = 0 in three-space.

3

Which axis is the plane 3y + 2z = 6 parallel to?

Normals and the Scalar Equation of a Plane

1

Find the plane through (2, −1, 0) with normal [1, 4, −2].

2

Find the plane through A(1, 1, 0), B(2, 0, 1) and C(0, 2, 2).

3

Find the acute angle between the planes x − y = 3 and y − z = 1.

Vector and Parametric Equations of a Plane

1

Find a scalar equation of r = [1, 0, 2] + s[1, 2, 0] + t[0, 1, −1].

2

Write a vector equation of x − y + 2z = 4.

3

Is (3, 4, 2) on r = [1, 1, 1] + s[1, 1, 0] + t[0, 1, 1]?

Two Planes and Their Line of Intersection

1

Find the line of intersection of x + y − z = 1 and 2x − y + z = 5.

2

Describe 2x − 4y + 6z = 3 and −x + 2y − 3z = 1.

3

Write the line r = [0, 2, 1] + t[1, 0, 3] as the intersection of two planes.

Intersections of Lines in Three-Space: Intersecting, Parallel and Skew

1

Classify r = [1, 2, 3] + t[2, −1, 1] and r = [3, 1, 5] + s[−4, 2, −2].

2

Classify r = [0, 1, 2] + t[1, 1, 0] and r = [2, 0, 0] + s[0, 1, 1].

3

Find the intersection of r = [1, 0, 1] + t[1, 1, 1] and r = [3, 1, 1] + s[0, 1, 2].

Intersection of a Line and a Plane

1

Where does x = 1 + 2t, y = −t, z = 3 + t meet 3x + y − z = 10?

2

Describe the intersection of r = [1, 1, 1] + t[1, 2, 3] with x + y − z = 1.

3

Find the angle between a line with direction [1, 1, 0] and the plane x = 0.

Three Planes: Geometric Configurations

1

Describe x + y + z = 2, x + y + z = 5 and x − z = 0.

2

Three planes have normals [1, 1, 0], [0, 1, 1] and [1, 0, 1]. Do they meet in a single point?

3

Each pair of three planes meets in a line, the lines are parallel and distinct, and there is no common point. Name the configuration.

Solving Systems of Three Planes Algebraically

1

Solve x + y + z = 4, x − y + z = 2, 2x + y − z = 1.

2

Solve x + y = 1, y + z = 2, x − z = −1.

3

Solve x + y = 1, y + z = 2, x − z = 4, and describe the configuration.

Distances from a Point to a Line and to a Plane

1

Find the distance from (2, 1, −1) to 2x − 2y + z = 9.

2

Find the distance from the origin to the line r = [1, 1, 0] + t[0, 0, 1].

3

Find the distance between 2x + y − 2z = 3 and 4x + 2y − 4z = −12.

Point to plane|Ax₀ + By₀ + Cz₀ + D|÷ √(A² + B² + C²)Point to line (3-D)|Q₀P × m|÷ |m|Skew lines|P₁P₂ · (m₁ × m₂)|÷ |m₁ × m₂|Each one is a length found by projection (dot product) or by area ÷ base (cross product).Distances are never negative: keep the absolute values.
The three distance formulas of the unit, side by side.

Distances Between Skew Lines, and Between a Line and a Plane

1

Find the distance between the y-axis and the line r = [4, 0, 1] + s[0, 0, 1].

2

Find the distance between r = [1, 0, 0] + t[1, 1, 0] and r = [0, 0, 1] + s[1, −1, 1].

3

Find the distance between the line r = [0, 1, 2] + t[1, 0, −1] and the plane x + y + z = 9.

Check your understanding

Quiz

1.

Which of these equations describes a plane in three-space?

2.

Which vector is a normal to the plane 4x − y + 3z = 2?

3.

Two planes have normals [1, 0, 0] and [0, 1, 0]. What is the direction of their line of intersection?

4.

The x-axis and the line r = [0, 1, 1] + s[0, 1, 0] are:

5.

Where does the line x = t, y = t, z = t meet the plane x + y + z = 9?

6.

Three planes have normals with n₁ · (n₂ × n₃) ≠ 0. Their intersection is:

7.

Elimination on three plane equations gives 0 = 0 exactly once, with one free variable. The planes meet in:

8.

What is the distance from the origin to the plane x + 2y + 2z = 6?

9.

Which is the scalar equation of the plane through the origin with normal [1, 1, 1]?

10.

A vector equation of a plane needs:

For parents and teachers

Use the review to find what to revisit:

  • Confusing coincident and parallel lines, or 0 = 0 with "no solution" → Lesson 1 (Lines and Linear Systems in Two-Space).
  • Using the two points as "point and direction", or different t-values when testing a point → Lesson 2 (Vector and Parametric Equations of a Line in Two-Space).
  • Mixing up the normal [A, B] and the direction [B, −A], or sign slips in symmetric form → Lesson 3 (Connecting Scalar, Vector, Parametric and Symmetric Equations of a Line).
  • Writing a "scalar equation of a line" in three-space, or dividing by a zero direction component → Lesson 4 (Vector, Parametric and Symmetric Equations of a Line in Three-Space).
  • Drawing y = x as a line in three-space, or intercept sign errors → Lesson 5 (Linear Equations in Three Variables: Planes).
  • Cross-product slips, or forgetting D when planes share a normal → Lesson 6 (Normals and the Scalar Equation of a Plane).
  • Using parallel direction vectors, or the normal as a direction → Lesson 7 (Vector and Parametric Equations of a Plane).
  • Expecting two planes to meet in a point, or giving up when z = 0 gives no point → Lesson 8 (Two Planes and Their Line of Intersection).
  • Not checking the third equation, or calling skew lines parallel → Lesson 9 (Intersections of Lines in Three-Space: Intersecting, Parallel and Skew).
  • Substituting into a plane still in vector form, or using cos instead of sin for the line-plane angle → Lesson 10 (Intersection of a Line and a Plane).
  • Thinking "no common point" means the planes are parallel → Lesson 11 (Three Planes: Geometric Configurations).
  • Elimination slips, or not describing the configuration after finding "no solution" → Lesson 12 (Solving Systems of Three Planes Algebraically).
  • Wrong sign of D in the distance formula, or a missing absolute value → Lesson 13 (Distances from a Point to a Line and to a Plane).
  • Using the skew-line formula on parallel lines → Lesson 14 (Distances Between Skew Lines, and Between a Line and a Plane).

Looking ahead: this is the final unit of MCV4U. Students heading to university in mathematics, engineering, physics, computer science or economics will meet these ideas again right away in linear algebra, where systems of many equations are solved with matrices, and in multivariable calculus, where planes become the "tangent planes" of curved surfaces.