Vectors: angle between vectors 1

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Bill Foster

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Bill Foster 55 years, 2 months ago

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Question statement

Given the vectors:
$\boldsymbol{A}=\simplify[std]{{s1}v:e_{a}+{s2}v:e_{b}},\;\;\;\boldsymbol{B}=\simplify[std]{{s3}v:e_{c}+{s4}v:e_{d}}$

Find the angle in degrees and radians between them.

Note the angle must be between $0\,^{\circ}$ and $180\,^{\circ}$ (between $0$ and $\pi$ radians)

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	1
b	integer	3
angle	number	120.0
d	integer	2
g	integer	-1
s3	integer	1
s2	integer	1
s1	integer	1
s4	integer	-1
fa	integer	0
fb	integer	1
u	integer	3
t	integer	1
sb	integer	-1
sa	integer	1
tb	integer	0
ta	integer	1

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Name

Data type

Value

xxxxxxxxxx
 
if(t=1,2,1)

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Description

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Generated value: `integer`

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Parts

Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Mathematical expression
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Gap-fill

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Angle in degrees = Gap 0. $^{\circ}$

Angle in radians = Gap 1.radians.

Note that you can input the radians as a decimal to 4 decimal places or as a multiple of $\pi$ . You input $\pi$ using pi.

Advice

Use the formula:

$\boldsymbol{A \cdot B} = |\boldsymbol{A}||\boldsymbol{B}|\cos(\theta)$ where $\theta$ is the angle between the vectors.

Here $|\boldsymbol{A}| = \sqrt{ (\var{s1})^2+(\var{s2})^2} = \simplify[all]{sqrt({s1^2+s2^2})},\;\;\;|\boldsymbol{B}| = \sqrt{ (\var{s3})^2+(\var{s4})^2} = \simplify[all]{sqrt({s3^2+s4^2})}$

and

$\boldsymbol{A \cdot B} = (\var{fa},\var{sa}, \var{ta}) \cdot (\var{fb},\var{sb}, \var{tb}) = \var{g}$ .

So $\begin{eqnarray*} \cos(\theta)&=&\frac{\var{g}}{\sqrt{2}\sqrt{2}} = \simplify[std]{{g}/{2}}\\ \Rightarrow \theta &=&\arccos\left(\simplify[std]{{g}/{2}}\right)\\ &=&\var{angle}\,^{\circ} \end{eqnarray*}$
Converting from degrees to radians is simply a matter of multiplying the angle in degrees by $\displaystyle \frac{\pi}{180}$ .

Hence $\displaystyle \var{angle}\,^{\circ}=\simplify[std]{({angle}*pi)/{180}= {precround(angle*pi/180,4)}}$ radians to 4 decimal places.

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Taxonomy: mathcentre

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Vectors: angle between vectors 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`