Equations of lines and planes

Question 1

What is the vector equation of any line L?

Answer 1

r=r₀+λv

Question 2

Answer 2

Question 3

What is the relationship between the vector equation and parametric representation?

Answer 3

The parametric representation is a scalar representation of the vector equation.

Question 4

Convert the following vector equation to parametric equation.

Answer 4

Basically split the components

Question 5

Answer 5

Question 6

How would you derive symmetric equatioins from the parametric equations?

Answer 6

Isolate the lamda and equate the equations

Question 7

In symmetric equations, what do you do when the denominator is 0?

Answer 7

Basically x is equal to x₀

Question 8

Answer 8

Question 9

Answer 9

Question 10

Given r=r₀+λv and r=s₀+λu

When are these two lines parallel

Answer 10

They are parallel if the direction fectors are linear multiples of each other

Question 11

Answer 11

Question 12

Answer 12

This is a horizontal plane

Question 13

Answer 13

This is a vertical plane

Question 14

What is the general equation of a plane in 3D space?

Answer 14

ax+by+cz=d

Where a, b, c, d are fixed real numbers

Question 15

Given the equation of a plane by: ax+by+cz=d

When is the plane vertical?

Answer 15

The plane is verticle when c=0. The plane means it only intersects the x and one point and y at one point. When c=0, there is no z and so it has no ‘angle’ to it and looks like the following image.

Question 16

Given the plane equation, ax+by+cz=d

How can this be written when it is not vertical?

Answer 16

mx+ny+z₀

Question 17

Answer 17

Question 18

True or false?

Multiplying any plane equation by a nonzero constant gives the same plane back.

Answer 18

True.

Just think it as like, scaling triangle up. It still has the same dimensions but scaled.

Question 19

Answer 19

Question 20

The following picture shows the intersection of the planes 2x + 3y − z + 6 = 0 and z = 9x − 4y + 5. Note that the intersection is a line (Just revision)

Question 21

What else is needed for a plane to be represented with a point?

Answer 21

A vector that is perpendicular to the plane

Question 22

What is the general vector equation for a plane?

Answer 22

0 = n ⋅ (r - r₀)

Note it is dot product between two vectors

Question 23

Prove that the vector equation of a plane and scalar equation of a plane are the same

Answer 23

Question 24

Given three points in R^3</sub>

When do these points determine a plane?

(What condition)

Answer 24

When all three points do no lie on a straight line.

Because no straight line can form with three points defining a plane in 3D. But if they are, it can form a plane in 2D or just a line.

Question 25

# Given the plane ax+by+cz=d What can be said about the vector (a, b, c)

Answer 25

It is normal/perpendicular to the plane. ## Footnote In the image, n=(a, b, c). And from the definition of vector equation of a plane, (a, b, c) is perpendicular/normal to the plane.

Question 26

Answer 26

Question 27

# True or false? The angle between two planes cannot exceed pi/2

Answer 27

True

Question 28

Answer 28

Question 29

What is the second vector equation of a plane (similar to vector equation of a line)

Answer 29

r=r₀+λu+μv | Where λ, μ ∈ ℝ

Question 30

# In the plane equation r=r₀+λu+μv Describe the relationship between u and v with each other. Then describe the the relationbetween u and v with the line

Answer 30

u and v are non-parallel to each other. However, u and v are parallel to the plane

Question 31

# Given the plane r=r₀+λu+μv Draw this plane and display the components of it.

Answer 31

Question 32

Answer 32

Question 33

Answer 33

Question 34

Answer 34

## Footnote Vector product is performed because it gives a vector that is perpendicular to both n₁ and n₂. This new vector is the line that intersects the planes.

Question 35

Answer 35

Question 36

Answer 36