Q1 angle between two straight lines

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Stephen Bowlzer

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Stephen Bowlzer 8 years, 8 months ago

Created this as a copy of Angle between vectors.

Name	Status	Author	Last Modified
Angle between vectors	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Q1 angle between two straight lines	draft	Stephen Bowlzer	22/06/2016 14:40
Q2 intersection of two straight lines	draft	Stephen Bowlzer	24/06/2016 15:29
Vectors 1	draft	Violeta CIT	10/11/2017 15:40
Simon's copy of Angle between vectors	draft	Simon Thomas	01/04/2019 10:21
Timur's copy of Angle between vectors	draft	Timur Zaripov	03/04/2019 11:18
Angle between vectors	draft	Xiaodan Leng	10/07/2019 21:14
Angle between vectors	draft	Xiaodan Leng	10/07/2019 23:18

There are 18 other versions that do you not have access to.

Question statement

You are given the straight lines $\,{l}_1: \boldsymbol{\mathrm{r}} = \var{v_a} + t\var{v_b}\,\,$ and $\,\,{l}_2: \boldsymbol{\mathrm{r}} = \var{v_c} + s\var{v_d}$

Give any introductory information the student needs.

			Name	Type	Generated Value

direction_a	list	List of 3 items
direction_b	list	List of 3 items
direction_c	list	List of 3 items
direction_d	list	List of 3 items
s1	integer	1
s2	integer	-1
s3	integer	-1
s4	integer	-1
s5	integer	-1
s6	integer	1
s7	integer	-1
s8	integer	-1
units	list	List of 3 items
v_a	vector	vector(5,-6,0)
v_b	vector	vector(-2,0,-2)
v_c	vector	vector(2,0,-5)
v_d	vector	vector(-3,0,-4)
len_vb	number	2.8284271247
len_vd	number	5
dot_vbd	number	14
len_vbd	number	14.1421356237
cos_vbd	number	0.9899494937

			Name	Type	Generated Value

			angle	number	0.1418970546
			precision	integer	3

			Name	Type	Generated Value

t	integer	3
u	integer	2
a	integer	1
b	integer	2
c	integer	1
d	integer	3
fa	integer	1
fb	integer	-1
g	integer	-1
sa	integer	-1
sb	integer	0
ta	integer	0
tb	integer	-1
sa1	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
map(units[j],j,shuffle(0..2))

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `list`

[ vector(1,0,0), vector(0,1,0), vector(0,0,1) ]

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Number entry
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Find the angle between the straight lines ${l}_1$ and ${l}_2$ , in radians.

Note the angle must be in the range $0$ to $\pi$ .

Give your answer to {precision} decimal places.

Angle in radians = Unnamed gap

Advice

Use the formula, $\boldsymbol{a \cdot b} = \lVert\boldsymbol{a}\rVert\!\lVert\boldsymbol{b}\rVert \cos(\theta)$ where $\theta$ is the angle between the vectors.

Here

$\begin{align} \lVert \boldsymbol{a} \rVert &= \simplify[]{sqrt({v_b[0]}^2 + {v_b[1]}^2 + {v_b[2]}^2)} &= \var{precround(len_vb,4)} \\[1em] \lVert \boldsymbol{b} \rVert &= \simplify[]{sqrt({v_d[0]}^2 + {v_d[1]}^2 +{v_d[2]}^2)} &= \var{precround(len_vd,4)} \\[1em] \boldsymbol{a \cdot b} &= \var{v_b} \boldsymbol{\cdot} \var{v_d} &= \var{dot(v_b,v_d)} \end{align}$

$\begin{align} \cos(\theta) &= \frac{\var{dot(v_b,v_d)}}{\var{precround(len_vb,4)}\times\var{precround(len_vd,4)}} = &\var{precround(cos_vbd,4)} \\[1em] \implies \theta &= \arccos\left(\var{precround(cos_vbd,4)}\right) \\[1em] &= \var{precround(angle,precision)} \text{ radians} \end{align}$

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Part	Test	Passed?
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This question is used in the following exam:

Stephen's copy of MATH6005 Vectors by Stephen Bowlzer in Stephen's workspace.

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Taxonomy: mathcentre

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Checkpoint description

Stephen Bowlzer 8 years, 8 months ago

Error in variable testing condition

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Generated value: `list`

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Q1 angle between two straight lines

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Stephen Bowlzer 8 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`