Coding randomised different scenarios of a multiple choice answer

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Julia Goedecke

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4 years, 7 months ago

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Julia Goedecke 4 years, 7 months ago

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Created this as a copy of Planes in

$\mathbb{R}^3$ (randomized).

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 whether the following is a plane. If it is a plane, 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 plane may not be written in the most efficient way, so you have to check explicitely whether it go through the origin $\begin{pmatrix}0\\0\\0\end{pmatrix}$ or not.

Give any introductory information the student needs.

			Name	Type	Generated Value

a0	vector	vector(5,1,4)
u0	vector	vector(1,-4,3)
lambda	integer	-3
v0parallel	vector	vector(-3,12,-9)
a1	vector	vector(-2,-2,-2)
u1lower	list	[ 3, 4 ]
u1	vector	vector(0,3,4)
v1upper	list	[ 3, 1 ]
v1	vector	vector(3,1,0)
a2	vector	vector(-12,-12,-9)
u2	vector	vector(8,8,7)
v2	vector	vector(4,4,3)
a3	vector	vector(-1,0,3)
u3	vector	vector(0,0,1)
v3	vector	vector(-1,0,2)
a	vector	vector(-12,-12,-9)
u	vector	vector(8,8,7)
v	vector	vector(4,4,3)
d	integer	2

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(repeat(random(1..6),3))

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `vector`

vector(5,1,4)

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$\left\{\var{a}+ \lambda \var{u} + \mu \var{v} \middle| \lambda, \mu \in \mathbb{R} \right\}$ 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, or spotting patterns. We will learn how to solve it in general in Chapter 2.

Clearly $\var{u}$ and $\var{v}$ are parallel: $\var{v}=\var{lambda}\var{u}$ . So this is not a plane.

Because of the $0$ entries, it is clear that $\var{u}$ and $\var{v}$ are not parallel. So this is a plane. If we try to get $\var{a}+\lambda \var{u}+\mu \var{v}=\begin{pmatrix}0\\0\\0\end{pmatrix}$ , we would need $\lambda = \simplify[fractionNumbers]{{-a[2]/u[2]}}$ to sort out the bottom row, and $\mu=\simplify[fractionNumbers]{{-a[0]/v[0]}}$ to sort out the top row, but this leaves the middle row non-zero: $\simplify[fractionNumbers]{{a}+{-a[2]/u[2]} {u}+{-a[0]/v[0]} {v}}= \simplify[fractionNumbers]{{a-a[2]/u[2]*u-a[0]/v[0]*v}}$ . So this plane does not go through the origin.

The third entry ensures that $\var{u}$ and $\var{v}$ are not parallel, so this is a plane. But since $\var{a}=\var{lambda}\var{v}$ , we see that $\lambda=0$ and $\mu=\var{-lambda}$ gives us the zero vector, so this plane does go through the origin.

The first entry ensures that $\var{u}$ and $\var{v}$ are not parallel, so this is a plane. In this case it is perhaps a bit more tricky to find if it goes through the origin. But if you spot that $\var{u}+\var{v}=\var{a}$ , then it is clear that $\lambda=\mu=-1$ gives the zero vector. So this plane goes through the origin.

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This question is used in the following exam:

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Coding randomised different scenarios of a multiple choice answer

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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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Coding randomised different scenarios of a multiple choice answer

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

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

Error

Warning

Generated value: vector

Parts

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Generated value: `vector`