Dot product - find angles between two pairs of vectors,

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Bill Foster

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Bill Foster 6 years, 6 months ago

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Name	Status	Author	Last Modified
Dot product - find angles between two pairs of vectors,	Has some problems	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Harry's copy of Dot product - find angles between two pairs of vectors,	draft	Harry Flynn	13/03/2018 11:42
Dot product and angle between two vectors	draft	Harry Flynn	13/03/2018 12:24
Vectors 2 Dot product and angle between two vectors	Needs to be tested	Violeta CIT	09/11/2019 17:58
Dot product and angle between two vectors	draft	Marie Nicholson	08/11/2018 12:52
MATH6005 Assessment 2_Q3of6	draft	Violeta CIT	12/11/2018 14:00
Simon's copy of Dot product and angle between two vectors	draft	Simon Thomas	01/04/2019 10:04

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Question statement

Find the cosine of the angle $\theta$ between the following pairs of vectors.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	vector	vector(-9,3,2)
lenb	number	13.3041346957
c	vector	vector(-3,2,-9,-5)
b	vector	vector(8,8,-7)
lenc	number	10.9087121146
d	vector	vector(-5,-3,-8,9)
lend	number	13.3790881603
ans1	number	-0.48
ans2	number	0.25
lena	number	9.6953597148

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(repeat(random(1..9)*sign(random(1,-1)),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(-9,3,2)

→ Used by:

ans1
lena

Advice
"Part a)" - prompt

Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Gap-fill
1. Gap Number entry
  Add

Gap-fill

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

$\boldsymbol{a}=\pmatrix{\var{a[0]},\var{a[1]},\var{a[2]}}$ and $\boldsymbol{b}=\pmatrix{\var{b[0]},\var{b[1]},\var{b[2]}}$

$\cos({\theta})=$ Unnamed gap. (Enter your answer to 2d.p.)

Advice

Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.

The dot product of two vectors $\boldsymbol{a}=\pmatrix{a_1,a_2,a_3}$ and $\boldsymbol{b}=\pmatrix{b_1,b_2,b_3}$ is given by

$\boldsymbol{a\cdot b}=a_1b_1+a_2b_2+a_3b_3.$

It is also given by

$\boldsymbol{a\cdot b}=ab\cos(\theta)$

where $a=\lvert\boldsymbol{a}\rvert=\sqrt{a_1^2+a_2^2+a_3^2}$ and $b=\lvert\boldsymbol{b}\rvert=\sqrt{b_1^2+b_2^2+b_3^2}$ are the lengths of the vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ .

Equating the two expressions gives

$a_1b_1+a_2b_2+a_3b_3=ab\cos(\theta)$

and so

$\cos(\theta)=\frac{a_1b_1+a_2b_2+a_3b_3}{ab}.$

In part a) therefore, we have

$\cos(\theta)=\frac{\simplify[std]{{a[0]*b[0]}+{a[1]*b[1]}+{a[2]*b[2]}}}{\var{precround(lena,2)}\times\var{precround(lenb,2)}}=\frac{\var{dot(a,b)}}{\var{precround(lena*lenb,2)}}=\var{ans1} \; \text{to 2d.p.,}$

and in part b) we have

$\cos(\theta)=\frac{\simplify[std]{{c[0]*d[0]}+{c[1]*d[1]}+{c[2]*d[2]}+{c[3]*d[3]}}}{\var{precround(lenc,2)}\times\var{precround(lend,2)}}=\frac{\var{dot(c,d)}}{\var{precround(lenc*lend,2)}}=\var{ans2} \; \text{to 2d.p.}$

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Dot product - find angles between two pairs of vectors,

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Bill Foster 6 years, 6 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: `vector`

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Gap-fill

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Dot product - find angles between two pairs of vectors,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 6 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: vector

Parts

Gap-fill

Generated value: `vector`