Find unit vector orthogonal to two others,

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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
Find unit vector orthogonal to two others,	Has some problems	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Simon's copy of Find unit vector orthogonal to two others,	draft	Simon Thomas	01/04/2019 10:34

Question statement

Find the unit vector $\boldsymbol{\hat{a}}$ , with positive $x$ -component, which is orthogonal to both $\boldsymbol{u}=\pmatrix{\var{u[0]},\var{u[1]},\var{u[2]}}$ and $\boldsymbol{v}=\pmatrix{\var{v[0]},\var{v[1]},\var{v[2]}}$ .

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			Name	Type	Generated Value

u	vector	vector(9,4,3)
v	vector	vector(4,-4,-8)
a	vector	vector(-20,84,-52)
apos	vector	vector(20,-84,52)
unitapos	vector	Vector with 3 components

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Name

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Value

xxxxxxxxxx
 
vector(repeat(random(-9..9 except 0),3))

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Generated value: `vector`

vector(9,4,3)

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$\boldsymbol{\hat{a}}=$ Unnamed gap (Enter your answers to 3d.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.

A vector $\boldsymbol{a}$ , which is orthogonal to both $\boldsymbol{u}$ and $\boldsymbol{v}$ , is given by

$\boldsymbol{u}\wedge\boldsymbol{v}=\pmatrix{u_2 v_3 - u_3 v_2, & u_3 v_1 - u_1 v_3, & u_1 v_2 - u_2 v_1}$

The magnitude of $\boldsymbol{a}$ is given by

$\lvert\boldsymbol{a}\rvert=\sqrt{a_1^2+a_2^2+a_3^2}$

A unit vector $\boldsymbol{\hat{a}}$ is obtained by dividing the components of the vector $\boldsymbol{a}$ by its magnitude, i.e.

$\boldsymbol{\hat{a}}=\frac{\boldsymbol{a}}{\lvert\boldsymbol{a}\rvert}$

In this question,

$\boldsymbol{a} = \pmatrix{\simplify[basic]{{u[1]}*{v[2]}-{u[2]}*{v[1]}}, & \simplify[basic]{{u[2]}*{v[0]}-{u[0]}*{v[2]}}, & \simplify[basic]{{u[0]}*{v[1]} - {u[1]}*{v[0]}}} = \var[rowvector]{a}$

and

$\lvert\boldsymbol{a}\rvert = \sqrt{(\var{a[0]})^2+(\var{a[1]})^2+(\var{a[2]})^2} = \var{precround(len(a),3)} \text{ to 3 decimal places.}$

The unit vector with positive $x$ -component is therefore $\boldsymbol{\hat{a}}=\frac{1}{\var{precround(len(a),3)}}\var[rowvector]{apos} = \pmatrix{\var{precround(unitapos[0],3)}, & \var{precround(unitapos[1],3)}, & \var{precround(unitapos[2],3)}}$ to 3d.p.

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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.
Workbook 2 by Jessica Murray in Jessica's workspace.
Vector calculus exam by Simon Thomas in Vector calculus.
Nicolas's copy of Vector calculus by Nicolas Sawyer in Nicolas's workspace.
Vector calculus by Michael Proudman in .Vectors.
Vector calculus by steve kilgallon in 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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Find unit vector orthogonal to two others,

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

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Generated value: vector

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Gap-fill

Generated value: `vector`