Elementary operations on vectors,

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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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	-4
q	matrix	matrix([-16,-14],[-14,-6])
s3	integer	-1
s2	integer	-1
s1	integer	-1
s5	integer	-1
s4	integer	-1
ssquares	number	329
v1	vector	vector(-8,-7,-9)
v2	vector	vector(-7,-3,7)
a4	integer	8
a	integer	-8
c	integer	-7
b	integer	-7
d	integer	-3
g	integer	-9
f	integer	7
m	matrix	matrix([-8,-7],[-7,-3])
n	matrix	matrix([-8,-7],[-7,-3])
ssquaresb	number	107
ssquaresa	number	194
v	vector	vector(-15,-10,-2)

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`

-4

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
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2. Gap Gap 1. Matrix entry
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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)

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

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

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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This question is used in the following exams:

Vector calculus by Newcastle University Mathematics and Statistics in Content created by Newcastle University.
Jorj's copy of Vector calculus by Jorj Kowszun in Jorj's workspace.
Alvaro's copy of Vector calculus by Alvaro Martínez in Alvaro's workspace.
Vector calculus by steve kilgallon in Vector Calculus.
Vector calculus exam by Simon Thomas in Vector calculus.
Vector calculus by Michael Proudman in .Vectors.
CEG2705 Assessment 1 by Ryan Doran in NCL CEG2705.
All the good stuff by Chris Templet in Chris's workspace.
Vector calculus by steve kilgallon in Vectors.
Vectors by Angeliki Karperaki in Angeliki's workspace.
vector by Tony Amparayil Francis in Tony's workspace.

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Bill Foster 6 years, 6 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Elementary operations on vectors,

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Taxonomy: mathcentre

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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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Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

d)

Generated value: `integer`