Harry's copy of Elementary operations on vectors,

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Harry Flynn

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Harry Flynn 7 years, 1 month ago

Created this as a copy of Elementary operations on vectors, .

Name	Status	Author	Last Modified
Elementary operations on vectors,	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Harry's copy of Elementary operations on vectors,	draft	Harry Flynn	09/03/2018 14:35
Laure's copy of Harry's copy of Elementary operations on vectors,	draft	Laure Helme-Guizon	28/06/2018 05:37
Operasi Vektor	draft	aleams barra	25/07/2018 02:30
Elementary operations on vectors	draft	Paul Emanuel	20/08/2018 01:33
Vectors 1 Elementary operations on vectors	Needs to be tested	Violeta CIT	17/10/2018 10:47
Elementary operations on vectors	draft	Marie Nicholson	08/11/2018 12:24
MATH6005 Assessment 2_Q1of6	draft	Violeta CIT	12/11/2018 15:46
MATH6005 Assessment 2_Q2of6	draft	Violeta CIT	06/11/2019 15:13
Simon's copy of Elementary operations on vectors	draft	Simon Thomas	01/04/2019 09:49
Maria's copy of Simon's copy of Elementary operations on vectors	draft	Maria Aneiros	24/05/2019 00:49
Elementary operations on vectors,	draft	Xiaodan Leng	10/07/2019 23:19

There are 21 other versions that do you not have access to.

Question statement

You are given the vectors

$\begin{align} \boldsymbol{v} & =\simplify[std]{vector({a},{b},{g})}, & \boldsymbol{w} &= \simplify[std]{vector({c},{d},{f})}\qquad \in{\mathbb R}^3. \end{align}$

Enter your answers to the following questions exactly, using the function sqrt(x) if necessary.

Give any introductory information the student needs.

			Name	Type	Generated Value

b4	integer	-6
q	matrix	matrix([-10,12],[-18,14])
s3	integer	-1
s2	integer	1
s1	integer	-1
s5	integer	-1
s4	integer	1
ssquares	number	374
v1	vector	vector(-5,6,-9)
v2	vector	vector(-9,7,6)
a4	integer	9
a	integer	-5
c	integer	-9
b	integer	6
d	integer	7
g	integer	-9
f	integer	6
m	matrix	matrix([-5,6],[-9,7])
n	matrix	matrix([-5,6],[-9,7])
ssquaresb	number	166
ssquaresa	number	142
v	vector	vector(-14,13,-3)

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
-random(3..9)

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: `integer`

-6

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Matrix entry
  Add
Part b) Gap-fill
Part c) Gap-fill
Part d) Gap-fill
1. Gap Gap 0. Matrix entry
  Add
2. Gap Gap 1. Matrix entry
  Add
Part e) Gap-fill

Gap-fill

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

Find $\boldsymbol{v}+\boldsymbol{w} =$ Unnamed gap

Advice

a)

$\boldsymbol{v}+\boldsymbol{w} = \var{vector(a,b,g)} + \var{vector(c,d,f)} = \var{vector(a+c,b+d,g+f)}$

b)

In general for a vector $\boldsymbol{x}= \begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix}$ , we have $\lVert \boldsymbol{x} \rVert = \sqrt{x_1^2+x_2^2+x_3^2}$ .

Hence:

$\begin{align} \lVert \boldsymbol{v} \rVert &= \sqrt{\var{a^2}+\var{b^2}+\var{g^2}} = \simplify[all]{ sqrt({a^2+b^2+g^2})} \\ \lVert \boldsymbol{w} \rVert &= \sqrt{\var{c^2}+\var{d^2}+\var{f^2}} = \simplify[all]{ sqrt({c^2+d^2+f^2})} \\ \lVert \boldsymbol{v+w} \rVert &= \sqrt{\var{(a+c)^2}+\var{(b+d)^2}+\var{(g+f)^2}} = \simplify[all]{ sqrt({(a+c)^2+(b+d)^2+(f+g)^2})} \end{align}$

c)

Given a vector $\boldsymbol{x}= \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$ , the unit vector parallel to $\boldsymbol{x}$ is given by:

$\boldsymbol{u_x} = \frac{1}{\lVert \boldsymbol{x} \rVert} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} \frac{x_1}{\lVert \boldsymbol{x} \rVert} \\ \frac{x_2}{\lVert \boldsymbol{x} \rVert} \\ \frac{x_3}{\lVert \boldsymbol{x} \rVert} \end{pmatrix}$

For this example we have $\lVert \boldsymbol{v+w} \rVert =\simplify[std]{sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}$ , hence:

$\begin{align} &&\boldsymbol{z} = \boldsymbol{v} + \boldsymbol{w} &= \begin{pmatrix} \var{a+c} \\ \var{b+d} \\ \var{g+f} \end{pmatrix} \\ \implies && \boldsymbol{u_z} &= \frac{1}{\sqrt{\var{ssquares}}} \begin{pmatrix} \var{a+c} \\ \var{b+d} \\ \var{g+f} \end{pmatrix} \\[1em] && &= \begin{pmatrix} \simplify[std]{{a+c}/sqrt({ssquares})} \\ \simplify[std]{{b+d}/sqrt({ssquares})} \\ \simplify[std]{{g+f}/sqrt({ssquares})} \end{pmatrix} \end{align}$

d)

$\begin{align} \var{a4}\boldsymbol{v} &= \simplify{vector({a4}*{a}, {a4}*{b}, {a4}*{g})} \\[1em] &= \var{a4*vector(a,b,g)} \end{align}$

$\begin{align} \var{-b4}\boldsymbol{v} &= \simplify{vector({-b4}*{c}, {-b4}*{d}, {-b4}*{f})} \\[1em] &= \var{-b4*vector(c,d,f)} \end{align}$

e)

Using the information above, the unit vector parallel to $\boldsymbol{v}$ is:

$\boldsymbol{u_v} = \begin{pmatrix} \simplify[std]{{a}/sqrt({ssquaresA})} \\ \simplify[std]{{b}/sqrt({ssquaresA})} \\ \simplify[std]{{g}/sqrt({ssquaresA})} \end{pmatrix}$

and the unit vector anti-parallel to $\boldsymbol{w}$ is:

$-\boldsymbol{u_w} = \begin{pmatrix} \simplify[std]{{-c}/sqrt({ssquaresB})} \\ \simplify[std]{{-d}/sqrt({ssquaresB})} \\ \simplify[std]{{-f}/sqrt({ssquaresB})} \end{pmatrix}$

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Harry's copy of Elementary operations on vectors,

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Taxonomy: mathcentre

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Contributors

History

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Checkpoint description

Harry Flynn 7 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

d)

e)

Loading...

Error

Harry's copy of Elementary operations on 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: integer

Parts

Gap-fill

a)

b)

c)

d)

e)

Generated value: `integer`