Unit 6- Lines and Planes

Lesson Package

MCV4U

&

Unit 6 Outline Unit Goal: By the end of this unit, you will be able to demonstrate an understanding of vectors in two-space

by representing them geometrically and by recognizing their applications.

Section Subject Learning Goals Curriculum

Expectations

L1 Vector Equation of a

Line - Be able to write the vector and scalar equation of a line in 2-space

C4.1, C4.2

L2 Equations of Lines in

𝑅3 - Be able to write the vector and parametric equation of a line in 3-space

C4.1, C4.2

L3 Equations of Planes - Be able to write vector and parametric equations of planes C4.5

L4 Equations of Planes in

Scalar Form - Be able to write the scalar equation of a plane C4.3, C4.4,

C4.5, C4.6

L5 Iintersection of Lines - Solve for points of intersections of lines in 2 and 3 space C3.1, C3.2, C3.3

L6 Intersection of Lines

and Planes - Solve for the intersection of a line with a plane

C3.1, C3.2, C3.3

L7 Intersection of Planes - Solve for the intersection of 2 planes or 3 planes. C3.1, C3.2, C3.3

Assessments F/A/O Ministry Code P/O/C KTAC Note Completion A P Practice Worksheet Completion

F/A P

Quiz – Equations of lines and planes

F P

PreTest Review F/A P Test – Lines and Planes

O C3.1, C3.2, C3.3, C4.1, C4.2, C4.3,

C4.4, C4.5, C4.6 P K(25%), T(25%), A(25%),

C(25%)

L1 – Equations of Lines in Two-Space Unit 6 MCV4U Jensen In two-space, a line can be defined many different types of equations:

i) Slope 𝑦-intercept form → 𝑦 = 𝑚𝑥 + 𝑏

ii) Standard form (scalar equation) → 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0

iii) Vector Equation → 𝑟 = 𝑟0⃗⃗⃗⃗ + 𝑡�⃗⃗⃗� or [𝑥, 𝑦] = [𝑥0, 𝑦0] + 𝑡[𝑚1, 𝑚2]

iv) Parametric Equation → {𝑥 = 𝑥0 + 𝑡𝑚1

𝑦 = 𝑦0 + 𝑡𝑚2

Part 1: Vector Equation of a Line in Two-Space In slope 𝑦-intercept form and standard form, we have equations that define all points (𝑥, 𝑦) on the line. A vector equation of a line is an equation that describes resultant vectors that start at the origin and end at a point on the line. In order to create these resultant vectors, we need a position vector that gives us a point on the line, 𝑟0⃗⃗⃗⃗ = [𝑥0, 𝑦0], and a direction vector parallel to the line, �⃗⃗⃗� = [𝑚1,𝑚2]. By adding the position and direction vectors together, we get a resultant vector, 𝑟, that has its tip on a point on the line. By multiplying the direction vector by 𝑡 (to get 𝑡�⃗⃗⃗�), the resultant vector, 𝑟, can have its tip at any point on the line. Summary of Vector Equation of a Line: 𝑟 = 𝑟0⃗⃗⃗⃗ + 𝑡�⃗⃗⃗� or [𝑥, 𝑦] = [𝑥0, 𝑦0] + 𝑡[𝑚1, 𝑚2] where

• 𝑡 ∈ ℝ • 𝑟 = [𝑥, 𝑦] is a position vector to any unknown point on the line

• 𝑟0⃗⃗⃗⃗ = [𝑥0, 𝑦0] is a position vector to any known point on the line

• �⃗⃗⃗� = [𝑚1,𝑚2] is a direction vector parallel to the line

y

x

P(x0,y0)

P(x,y)L

Example 1: For a line that goes through the points 𝐴(1,4) and 𝐵(3,1) a) Write a vector equation for the line b) Determine three more position vectors to points on the line. Graph the line. https://www.geogebra.org/graphing/j6ntqajv

c) Determine if the point (2,3) is on the line. Part 2: Parametric Equation of a Line in Two-Space Remember that the Vector Equation of a line can be written as:

i) 𝑟 = 𝑟0⃗⃗⃗⃗ + 𝑡�⃗⃗⃗� or [𝑥, 𝑦] = [𝑥0, 𝑦0] + 𝑡[𝑚1, 𝑚2]

The vector equation of the line can be separated in to two parts, one for each variable. The parametric equations of a line in two-space are:

𝑥 = 𝑥0 + 𝑡𝑚1 𝑦 = 𝑦0 + 𝑡𝑚2

The parametric equation of the line from example 1 would be:

Example 2: Consider the line ℓ1: {𝑥 = 3 + 2𝑡𝑦 = −5 + 4𝑡

a) Find the coordinates of two points on the line. b) Write a vector equation of the line. c) Write the scalar equation of the line. To write the scalar equation, isolate 𝑡 in both parametric equations, then set them equal to each other and rearrange in to standard form.

d) Determine if ℓ1 is parallel to ℓ2: {𝑥 = 1 + 3𝑡𝑦 = 8 + 12𝑡

Example 3: Consider the line with scalar equation 4𝑥 + 5𝑦 + 20 = 0 a) Graph the line

b) Determine a position vector that is perpendicular to the line. c) How does the position vector from part b) compare to the scalar equation? d) Write a vector equation of the line

Example 4: Find a scalar equation for the line [𝑥, 𝑦] = [0,3] + 𝑡[2,−1]

Remember shortcut for finding a perpendicular vector is to swap coordinates and change 1 sign.

L2 – Equations of Lines in Three-Space Unit 6 MCV4U Jensen https://www.geogebra.org/3d/aku6q7xs In two-space lines can be represented using: vector equations, parametric equations, scalar equations, or an equation in slope 𝑦-intercept form. A line in three space can be defined by a vector, parametric, or symmetric equation but not a scalar equation. In three-space, a scalar equation defines a plane. A plane is a two-dimensional flat surface that extends infinitely in all directions. As in two-space, a line in three-space needs a position vector to a known point on the line and a direction vector in order to define it.

Equations of a line in 𝑹𝟑:

i) Vector Equation → 𝑟 = 𝑟0⃗⃗⃗⃗ + 𝑡�⃗⃗⃗� or [𝑥, 𝑦, 𝑧] = [𝑥0, 𝑦0, 𝑧0] + 𝑡[𝑚1,𝑚2, 𝑚3]

ii) Parametric Equation → {

𝑥 = 𝑥0 + 𝑡𝑚1

𝑦 = 𝑦0 + 𝑡𝑚2

𝑧 = 𝑧0 + 𝑡𝑚3

where 𝑟0⃗⃗⃗⃗ is the position vector and �⃗⃗⃗� is the direction vector.

iii) Symmetric Equation → 𝑥−𝑥0

𝑚1=

𝑦−𝑦0

𝑚2=

𝑧−𝑧0

𝑚3

The symmetric equation is derived from the parametric equations and solving for the 𝑡 parameter in each component.

Example 1: A line passes through points 𝐴(2,−1,5) and 𝐵(3,6,−4). a) Write a vector equation of the line.

b) Write parametric equations for the line. c) Determine if the point 𝐶(0, −15,9) lies on the line. Example 2: Find Vector, Parametric, and Symmetric equations of a line that passes through points 𝐴(2, −1,3) and 𝐵(5,1,1).

Notice that you cannot write a scalar equation of a line in 3-space. This is reserved for defining planes which we will do next lesson.

L3 – Equations of Planes Unit 6 MCV4U Jensen To determine the equation of a plane you need to know two non-collinear vectors parallel to the plane and a point on the plane. We can determine this information if any of the following information is given: The equation of a plane can be written in the following forms:

i) Vector ii) Parametric iii) Scalar (next lesson)

Vector Equation of a Plane Parametric Equation of a Plane

𝑟 = 𝑟0⃗⃗⃗⃗ + 𝑡�⃗� + 𝑠�⃗⃗�

OR [𝑥, 𝑦, 𝑧] = [𝑥0, 𝑦0, 𝑧0] + 𝑡[𝑎1, 𝑎2, 𝑎3] + 𝑠[𝑏1, 𝑏2, 𝑏3]

𝑟 = [𝑥, 𝑦, 𝑧] is a position vector for any point on the plane. 𝑟0⃗⃗⃗⃗ = [𝑥0, 𝑦0, 𝑧0] is a position vector for a known point on the plane.

�⃗� = [𝑎1, 𝑎2, 𝑎3] and �⃗⃗� = [𝑏1, 𝑏2, 𝑏3] are non-parallel direction vectors parallel to the plane. 𝑡 and 𝑠 are scalars

{

𝑥 = 𝑥0 + 𝑡𝑎1 + 𝑠𝑏1𝑦 = 𝑦0 + 𝑡𝑎2 + 𝑠𝑏2𝑧 = 𝑧0 + 𝑡𝑎2 + 𝑠𝑏2

Example 1: Consider the plane with direction vectors �⃗� = [8,−5,4] and �⃗⃗� = [1,−3,−2] through 𝑃0(3,7,0). a) Write the vector and parametric equations of the plane. b) Determine if the point 𝑄(−10,8,−6) is on the plane. If the point is on the plane, then there exists a single set of 𝑡 and 𝑠 values that satisfy the equations.

c) Find the coordinates of two other points on the plane. d) Find the 𝑥-intercept of the plane.

Example 2: Find the vector and parametric equations of each plane. a) The plane with 𝑥-intercept = 2, 𝑦-intercept = 4, and 𝑧-intercept = 5. 3D visualization: https://www.geogebra.org/3d/nsj2rkzf

b) The plane containing the line [𝑥, 𝑦, 𝑧] = [0,3,−5] + 𝑡[6,−2,−1] and parallel to the line [𝑥, 𝑦, 𝑧] =[1,7,−4] + 𝑠[1,−3,3].

L4 – Equations of Planes in Scalar Form Unit 6 MCV4U Jensen The scalar equation of a plane in three-space is 𝐴𝑥 + 𝐵𝑦 + 𝐶𝑧 + 𝐷 = 0, where �⃗� = [𝐴, 𝐵, 𝐶] is a normal vector to the plane. Example 1a: Determine the scalar equation of a plane that has normal vector �⃗� = [3,−2,5] and contains the point 𝑃0(1,2,−3).

Method 1: Dot Product

Method 2: Use 𝑨𝒙 + 𝑩𝒚 + 𝑪𝒛 + 𝑫 = 𝟎

Example 1b: Is vector 𝑎 = [4,1,−2] parallel to the plane?

Example 1c: Is vector �⃗� = [15,−10,25] normal to the plane? Example 2: Find the scalar equation of the plane containing the points 𝐴(−3,−1, −2), 𝐵(4,6,2), and 𝐶(5,−4,1).

https://www.geogebra.org/3d/u3n69tvc Example 3: Determine the scalar equation of each plane a) parallel to the 𝑥𝑧-plane; through the point (-7,8,9) Geogebra 3D visualization

b) containing the line [𝑥, 𝑦, 𝑧] = [1,2,4] + 𝑡[4,1,11] and perpendicular to [𝑥, 𝑦, 𝑧] = [4,15,8] + 𝑠[2,3,−1].

L5 – Intersections of Lines in 2-Space and 3-Space Unit 6 MCV4U Jensen Part 1: Intersection of Lines in 2-Space Possibilities for intersection of lines in 2-Space:

Intersections Graph Equations

1 intersection at a point

𝑦 = 2𝑥 + 3

𝑦 = −2𝑥 − 1

Infinite Points of Intersection (parallel and coincident)

𝑦 = 2𝑥 + 3 𝑦 = 2𝑥 + 3

No points of intersection (parallel and distinct)

𝑦 = 2𝑥 + 3 𝑦 = 2𝑥 − 1

Note: Linear systems in 2-space can be solved using substitution or elimination.

Example 1: Find the solutions of each system if they exist. a) ℓ1: 4𝑥 − 6𝑦 = −10 ℓ2: 6𝑥 − 9𝑦 = −15 b) ℓ1: [𝑥, 𝑦] = [1,5] + 𝑠[−6,8] ℓ2: [𝑥, 𝑦] = [2,1] + 𝑡[9, −12]

c) ℓ1: [𝑥, 𝑦] = [3,4] + 𝑠[3, −2] ℓ2: [𝑥, 𝑦] = [−2, −2] + 𝑡[3,5]

Note: You could at this point solve for 𝑠 and make sure you get the same solution. This step will be necessary when solving systems in 3-space.

Part 2: Intersection of Lines in 3-Space Possibilities for intersection of lines in 3-Space:

Intersections Graph Equations

No intersections

(parallel and distinct)

�⃗� = [−1,4,2] + 𝑠[−2,2,2]

�⃗⃗� = [1,3,1] + 𝑡[−2,2,2]

No intersections

(not parallel but skewed)

�⃗� = [−3, −2,4] + 𝑠[1,1,1]

�⃗⃗� = [2,5,4] + 𝑡[−2,4, −1]

1 intersection

(not parallel but on same plane)

�⃗� = [−3, −2,4] + 𝑠[1,1,1]

�⃗⃗� = [−3, −2,4] + 𝑡[−2,4, −1]

Infinite points of intersection

(parallel and coincident)

�⃗� = [−1,4, −2] + 𝑠[−2,1, −3]

�⃗⃗� = [1,3,1] + 𝑡[−2,1, −3]

Steps for solving the intersection of lines in 3-space: The first step is always to check if direction vectors are parallel or not by comparing direction vectors and seeing if one is a scalar multiple of the other (𝑚1⃗⃗ ⃗⃗ ⃗⃗ = 𝑘𝑚2⃗⃗ ⃗⃗ ⃗⃗ )

a. If yes, there could be infinite solutions OR no solutions. Test to see if a point is on both lines or just solve the system and see if you get no solutions or infinite solutions.

b. If no, they could intersect at a point OR they could be skewed and never intersect. Try solving the system and see if you can find parameters that result in the equations being equal. If you can, plug either parameter in to an original line and solve for the point of intersection.

Example 2: Determine if the lines intersect. If they do, find the coordinates of the point of intersection. a) ℓ1: [𝑥, 𝑦, 𝑧] = [5,11,2] + 𝑠[1,5, −2] ℓ2: [𝑥, 𝑦, 𝑧] = [1, −9,9] + 𝑡[2,10, −4]

Step 1: Check if lines are parallel.

Step 2: Test to see if (5,11,2) is on ℓ2. If it is on both lines, then the lines are coincident (infinite solutions). If it is not on both lines, the lines are parallel and distinct (no solutions)

b) ℓ1: [𝑥, 𝑦, 𝑧] = [7,2, −6] + 𝑠[2,1, −3] ℓ2: [𝑥, 𝑦, 𝑧] = [3,9,13] + 𝑡[1,5,5]

https://www.geogebra.org/3d/esvcf8kf

c) ℓ1: [𝑥, 𝑦, 𝑧] = [5, −4, −2] + 𝑠[1,2,3] ℓ2: [𝑥, 𝑦, 𝑧] = [2,0,1] + 𝑡[2, −1, −1]

https://www.geogebra.org/3d/jtjbgq55

Part 3: Distance Between Two Skew Lines The shortest distance between skew lines is the length of the common perpendicular. It can be calculated

using the formula 𝑑 = |𝑃1𝑃2⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗∙�⃗⃗�

|�⃗⃗�||, where 𝑃1 and 𝑃2 are any points on each line and �⃗⃗� = �⃗⃗⃗�1 × �⃗⃗⃗�2 is a normal

common to both lines. Example 3: Determine the distance between the skew lines. ℓ1: [𝑥, 𝑦, 𝑧] = [5, −4, −2] + 𝑠[1,2,3] ℓ2: [𝑥, 𝑦, 𝑧] = [2,0,1] + 𝑡[2, −1, −1]

https://www.geogebra.org/3d/sj5mmwuh

Notice this comes from the projection formula learned in the previous unit:

|𝑝𝑟𝑜𝑗 �⃗⃗� �⃗�| = |�⃗� ∙ �⃗⃗�

|�⃗⃗�||

L6 – Intersection of Lines and Planes Unit 6 MCV4U Jensen Part 1: Intersections of Lines and Planes

Given a line and plane in 3-space, there are three possibilities for the intersection of the line with the plane:

Intersect at 1 Point Infinite Number of Solutions No Solutions

The line lies on the plane

The line is parallel and distinct from the plane

Example 1: In each case, determine if the line and the plane intersect. If so, determine the solution. a) 𝜋1: 9𝑥 + 13𝑦 − 2𝑧 = 29

ℓ1: {𝑥 = 5 + 2𝑡

𝑦 = −5 − 5𝑡𝑧 = 2 + 3𝑡

https://www.geogebra.org/3d/a9j3u5u2

b) 𝜋2: 𝑥 + 3𝑦 − 4𝑧 = 10

ℓ2: {𝑥 = 4 + 6𝑡

𝑦 = −7 + 2𝑡𝑧 = 1 + 3𝑡

https://www.geogebra.org/3d/quagawhg c) 𝜋3: 4𝑥 − 𝑦 + 11𝑧 = −1 ℓ3: [𝑥, 𝑦, 𝑧] = [−2,4,1] + 𝑡[3,1, −1] https://www.geogebra.org/3d/emxyfbbx

Example 2: Without solving, determine if each line intersects the plane. a) 𝜋1: 3𝑥 − 𝑦 + 𝑧 = −6 ℓ1: [𝑥, 𝑦, 𝑧] = [2, −5,3] + 𝑠[3,2,1]

b) 𝜋2: 4𝑥 − 2𝑧 = 11 ℓ2: [𝑥, 𝑦, 𝑧] = [1,0,1] + 𝑠[−2,1, −4] Part 2: Distance from a Point to a Plane

Remember the formula for the magnitude of the projection of �⃗� on to �⃗⃗� is |𝑝𝑟𝑜𝑗 �⃗⃗� �⃗�| =|�⃗⃗�∙�⃗⃗�|

|�⃗⃗�|

The shortest distance between a point and a plane is the perpendicular distance. This distance, 𝑑, between a

point 𝑃 and a plane is 𝑑 = |𝑝𝑟𝑜𝑗 �⃗⃗� 𝑃𝑄⃗⃗ ⃗⃗ ⃗⃗ | =|𝑃𝑄⃗⃗ ⃗⃗ ⃗⃗ ∙�⃗⃗�|

|�⃗⃗�| where 𝑄 is ANY point on the plane and �⃗⃗� is a normal vector to

the plane.

Example 3: Find the distance between the plan 4𝑥 + 2𝑦 + 𝑧 − 16 = 0 and the point 𝑃(10,3, −8).

https://www.geogebra.org/3d/r5qebzz4

L7 – Intersection of Planes Unit 6 MCV4U Jensen Part 1: Intersection of 2 Planes

Line of Intersection Infinite Solutions No Solutions If 2 distinct planes intersect, the solution is the set of points that lie on the line of intersection.

If the planes are coincident, every point on the plane is a solution

Parallel and distinct planes do not intersect

When solving a system of 2 planes, check if the planes are parallel by analyzing their normal vectors. If they are parallel, determine if they are coincident (infinite solutions) or distinct (no solutions). If the normals are not parallel, that means the planes are not parallel either. You can find the line of intersection by:

i) Eliminating a variable using the method of elimination ii) Choose 1 of the remaining variables to be the parameter 𝑡 iii) Write the other 2 variables in terms of the parameter 𝑡 as well to get the parametric equation of

the line of intersection Example 1: Describe how the planes in each pair intersect. a) 𝜋1: 2𝑥 − 𝑦 + 𝑧 − 1 = 0 𝜋2: 𝑥 + 𝑦 + 𝑧 − 6 = 0

https://www.geogebra.org/3d/jbesfzd5

b) 𝜋3: 2𝑥 − 6𝑦 + 4𝑧 − 7 = 0 𝜋4: 3𝑥 − 9𝑦 + 6𝑧 − 2 = 0

https://www.geogebra.org/3d/pgsp8f68

c) 𝜋5: 𝑥 + 𝑦 − 2𝑧 + 2 = 0 𝜋6: 2𝑥 + 2𝑦 − 4𝑧 + 4 = 0 Part 2: Intersection of 3 Planes A system of three planes is consistent if it has one or more solutions. A system of three planes is inconsistent if it has no solution. 3 scenarios of consistent solutions:

Intersect at 1 Point Intersect in a Line Infinite Solutions

Normals are not parallel and not coplanar.

Normals are not parallel but they are coplanar.

Normals are parallel and equations are scalar multiples of eachother.

Normals are parallel if each one is a scalar multiple of the others. Normals are coplanar if �⃗� 1 ∙ �⃗� 2 × �⃗� 3 = 0

Example 2: For each set of planes, describe the number of solutions and how the planes intersect. a) 𝜋1: 2𝑥 + 𝑦 + 6𝑧 − 7 = 0 𝜋2: 3𝑥 + 4𝑦 + 3𝑧 + 8 = 0 𝜋3: 𝑥 − 2𝑦 − 4𝑧 − 9 = 0

https://www.geogebra.org/3d/mudhuneu

b) 𝜋4: 𝑥 − 5𝑦 + 2𝑧 − 10 = 0 𝜋5: 𝑥 + 7𝑦 − 2𝑧 + 6 = 0 𝜋6: 8𝑥 + 5𝑦 + 𝑧 − 20 = 0

https://www.geogebra.org/3d/rvmqunqn

3 scenarios of inconsistent solutions:

2 planes are parallel and the third intersects both of the parallel

planes

The planes intersect in pairs The planes are parallel and at least two are distinct

Two normals are parallel but the third is not

Normals are not parallel but they are coplanar.

Normals are parallel but the equations are not scalar multiples of each other

a) 𝜋1: 3𝑥 + 𝑦 − 2𝑧 = 12 𝜋2: 3𝑥 − 5𝑦 + 𝑧 = 8 𝜋3: 12𝑥 + 4𝑦 − 8𝑧 = −4 https://www.geogebra.org/3d/kd3vjrbb

b) 𝜋4: 𝑥 + 3𝑦 − 𝑧 = −10 𝜋5: 2𝑥 + 𝑦 + 𝑧 = 8 𝜋6: 𝑥 − 2𝑦 + 2𝑧 = −4 https://www.geogebra.org/3d/rnycrtv3

c) 𝜋7: 4𝑥 − 2𝑦 + 6𝑧 = 35 𝜋8: −10𝑥 + 5𝑦 − 15𝑧 = 20 𝜋9: 6𝑥 − 3𝑦 + 9𝑧 = −50

https://www.geogebra.org/3d/r3ysp3gm