Planes in $\mathbb{R}^3$ (same as WB Q1.6)

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Name	Status	Author	Last Modified
Planes in $\mathbb{R}^3$ (same as WB Q1.6)	Ready to use	Julia Goedecke	25/08/2021 12:57
Planes in $\mathbb{R}^3$ (randomized)	Ready to use	Julia Goedecke	25/08/2021 12:57
Coding randomised different scenarios of a multiple choice answer	Ready to use	Julia Goedecke	25/08/2021 12:57

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Question statement

Determine which of the following are planes.

For each plane, also say whether it goes through the origin or not.

CAREFUL: while we said that if the plane (or line) goes through the origin, we don’t use the fixed first vector (the one that does not have a parameter in front of it), the following planes may not be written in the most efficient way, so you have to check explicitely whether they go through the origin $\begin{pmatrix}0\\0\\0\end{pmatrix}$ or not.

Give any introductory information the student needs.

			Name	Type	Generated Value

u1	vector	vector(1,0,4)
v1	vector	vector(8,2,0)
a2	vector	vector(1,1,1)
u2	vector	vector(1,2,0)
v2	vector	vector(0,4,2)
a3	vector	vector(3,1,1)
u3	vector	vector(1,0,3)
v3	vector	vector(-2,0,-6)
a4	vector	vector(1,2,1)
u4	vector	vector(3,6,2)
v4	vector	vector(1,2,1)
a5	vector	vector(-1,3,0.5)
u5	vector	vector(-2,6,2)
v5	vector	vector(1,-3,-1.5)

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Name

Data type

Value

xxxxxxxxxx
 
vector(1,0,4)

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Description

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Generated value: `vector`

vector(1,0,4)

This variable doesn't seem to be used anywhere.

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$\left\{ \lambda \var{u1} + \mu \var{v1} \middle| \lambda, \mu \in \mathbb{R} \right\}$ Unnamed gap

Advice

Given $\left\{ a+\lambda u + \mu v \middle| \lambda,\mu \in \mathbb{R}\right\}$ , we can tell whether this is a plane or not by checking whether the vectors $u$ and $v$ are parallel: if they are parallel, then it is just a line, if they are not parallel, then it is a plane.

To check whether the plane goes through the origin or not, we have to work out whether we can pick any values for $\lambda$ and $\mu$ that make the whole expression $a+\lambda u + \mu v =0$ . For most cases, this can be done by making use of the $0$ entries. We will learn how to solve it in general in Chapter 2.

a) Because of the $0$ entries, it is clear that $\var{u1}$ and $\var{v1}$ are not parallel. So this is a plane. Setting $\lambda=\mu=0$ we see that we get $\begin{pmatrix}0\\0\\0\end{pmatrix}$ , so this plane goes through the origin.

b) Because of the $0$ entries, it is clear that $\var{u2}$ and $\var{v2}$ are not parallel. So this is a plane. If we try to get $\var{a2}+\lambda \var{u2}+\mu \var{v2}=\begin{pmatrix}0\\0\\0\end{pmatrix}$ , we would need $\lambda = -1$ to sort out the top row, and $\mu=-\frac{1}{2}$ to sort out the bottom row, but this leaves the middle row non-zero. So this plane does not go through the origin.

c) Here $\var{u3}$ and $\var{v3}$ are parallel: $\var{v3}=-2\var{u3}$ . So this is not a plane.

d) The third entry ensures that $\var{u4}$ and $\var{v4}$ are not parallel, so this is a plane. But we can easily see that $\lambda=0$ and $\mu=-1$ gives us the zero vector, so this plane does go through the origin.

e) Again the third entry ensures that $\var{u5}$ and $\var[fractionNumbers]{v5}$ are not parallel, so this is a plane. This time it is perhaps a bit more tricky to find if it goes through the origin. But if you spot that $\var{u5}+\var[fractionNumbers]{v5}=\var[fractionNumbers]{a5}$ , then it is clear that $\lambda=\mu=-1$ gives the zero vector. So this plane also goes through the origin.

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This question is used in the following exams:

Testing by Tim Moroney in Tim's workspace.
All the good stuff by Chris Templet in Chris's workspace.

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Planes in $\mathbb{R}^3$ (same as WB Q1.6)

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Taxonomy: mathcentre

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Julia Goedecke 4 years, 6 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

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Generated value: `vector`

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Planes in R3\mathbb{R}^3 (same as WB Q1.6)

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Feedback

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Checkpoint description

Julia Goedecke 4 years, 6 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Error in variable testing condition

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Generated value: vector

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Gap-fill

Planes in $\mathbb{R}^3$ (same as WB Q1.6)

Generated value: `vector`