Dot product and angle between two vectors

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Harry Flynn

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Harry Flynn 7 years, 1 month ago

Created this as a copy of Harry's copy of Dot product - find angles between two pairs of vectors, .

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

There are 7 other versions that do you not have access to.

Question statement

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

Give any introductory information the student needs.

			Name	Type	Generated Value

a	vector	vector(-1,-1,-2)
lenb	number	12.0830459736
c	vector	vector(5,8,6)
b	vector	vector(-9,-4,7)
lenc	number	11.1803398875
d	vector	vector(8,2,9)
lend	number	12.2065556157
ans1	number	-0.03
ans2	number	0.81
lena	number	2.4494897428
ansrad	number	1.6
ansrad2	number	0.6
dot_of_ab	number	-1
dot_of_cd	number	110

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(-1,-1,-2)

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
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3. Gap Gap 2. Mathematical expression
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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 $\boldsymbol{b}=\pmatrix{\var{b[0]},\var{b[1]},\var{b[2]}}$

$\boldsymbol{a} \cdot \boldsymbol{b}=$ Gap 2.

$\cos({\theta})=$ Gap 0. (Give your answer to 2d.p.)

$\theta=$ Gap 1.(Give your answer, in radians, to 1d.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

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

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

and so

$\cos(\theta)=\frac{\boldsymbol{a\cdot b}}{\lvert\boldsymbol{a}\rvert \lvert\boldsymbol{b}\rvert}=\frac{a_1b_1+a_2b_2+a_3b_3}{\lvert\boldsymbol{a}\rvert \lvert\boldsymbol{b}\rvert}.$

In part a) therefore, we have

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

Which gives an angle $\theta =\var{ansrad}$ radians to 1 d.p.

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This question is used in the following exams:

Dot product and angle between two vectors by Harry Flynn in Harry's workspace.
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Taxonomy: mathcentre

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Harry Flynn 7 years, 1 month ago

Error in variable testing condition

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Generated value: `vector`

Parts

Gap-fill

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Dot product and angle between two vectors

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Harry Flynn 7 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: vector

Parts

Gap-fill

Generated value: `vector`