7. Lägesvektorer - Versionshistorik

Tek: flyttade 7. Position Vectors till 7. Lägesvektorer

2010-03-01T14:01:57Z

flyttade 7. Position Vectors till 7. Lägesvektorer

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Versionen från 1 mars 2010 kl. 14.01

Ian den 11 februari 2010 kl. 12.02

2010-02-11T12:02:40Z

← Äldre version		Versionen från 11 februari 2010 kl. 12.02
Rad 36:		Rad 36:
	'''Solution'''		'''Solution'''

-	The tables below show the positions of the children at 1 second intervals for	+	The tables below show the positions in metres of the children at 1 second intervals for
	<math>0\le t\le 4</math>		<math>0\le t\le 4</math>
	.		.

Ian den 9 februari 2010 kl. 12.14

2010-02-09T12:14:49Z

← Äldre version		Versionen från 9 februari 2010 kl. 12.14
Rad 8:		Rad 8:


-	~~7. Position Vectors~~	+	== '''Key Points''' ==
-		+
-	Key Points	+

	Velocities can be expressed as vectors, for example		Velocities can be expressed as vectors, for example

	<math>\mathbf{v}=V\cos \alpha \mathbf{i}+V\sin \alpha \mathbf{j}</math>		<math>\mathbf{v}=V\cos \alpha \mathbf{i}+V\sin \alpha \mathbf{j}</math>
-

	When moving with a constant velocity		When moving with a constant velocity
Rad 24:		Rad 21:

	'''[[Example 7.1]]'''		'''[[Example 7.1]]'''
-	Two children, A and B, start at the origin and run so that their position vectors in metres at time t seconds are given by:	+
		+	Two children, A and B, start at the origin and run so that their position vectors in metres at time <math>t</math> seconds are given by:

-
	<math>{{\mathbf{r}}_{A}}=2t\mathbf{i}+t\mathbf{j}</math>		<math>{{\mathbf{r}}_{A}}=2t\mathbf{i}+t\mathbf{j}</math>
	and		and
	<math>{{\mathbf{r}}_{B}}=(6t-{{t}^{2}})\mathbf{i}+t\mathbf{j}</math>		<math>{{\mathbf{r}}_{B}}=(6t-{{t}^{2}})\mathbf{i}+t\mathbf{j}</math>
-

	Plot the paths of the two children for		Plot the paths of the two children for
Rad 39:		Rad 35:

	'''Solution'''		'''Solution'''
		+
	The tables below show the positions of the children at 1 second intervals for		The tables below show the positions of the children at 1 second intervals for
	<math>0\le t\le 4</math>		<math>0\le t\le 4</math>
Rad 77:		Rad 74:

	'''[[Example 7.2]]'''		'''[[Example 7.2]]'''
		+
	A boat moves so that at time t its position vector is r, where		A boat moves so that at time t its position vector is r, where

	<math>\mathbf{r}=(9t-180)\mathbf{i}+(5t-450)\mathbf{j}</math>		<math>\mathbf{r}=(9t-180)\mathbf{i}+(5t-450)\mathbf{j}</math>

-	and i and j are unit vectors directed due east and north respectively.	+	and <math>\mathbf{i}</math> and <math>\mathbf{j}</math> are unit vectors directed due east and north respectively.
		+
		+	a) Find the time when the boat is due east of the origin.
		+
		+	b) Find the time when it is due south of the origin.

-	~~(a) Find the time when the boat is due east of the origin.~~	+	c) Find the position of the boat when it is south east of the origin.
-	~~(b) Find the time when it is due south of the origin.~~	+
-	(c) Find the position of the boat when it is south east of the origin.	+

	'''Solution'''		'''Solution'''
-	(a) When the boat is due east of the origin the position vector will contain only	+
		+	a) When the boat is due east of the origin the position vector will contain only
	<math>\mathbf{i}</math>		<math>\mathbf{i}</math>
	terms and no		terms and no
	<math>\mathbf{j}</math>		<math>\mathbf{j}</math>
	terms.		terms.
-		+
-		+
	<math>\begin{align}		<math>\begin{align}
	& 5t-450=0 \\		& 5t-450=0 \\
Rad 100:		Rad 100:
	\end{align}</math>		\end{align}</math>

-		+	b) When the boat is due south of the origin the position vector will contain only
-	(b) When the boat is due south of the origin the position vector will contain only	+
	<math>\mathbf{j}</math>		<math>\mathbf{j}</math>
	terms and no		terms and no
	<math>\mathbf{i}</math>		<math>\mathbf{i}</math>
	terms.		terms.
-

	<math>\begin{align}		<math>\begin{align}
Rad 113:		Rad 111:
	\end{align}</math>		\end{align}</math>

-		+	c) When the boat is south east of the origin, its position vector will be of the form
-	(c) When the boat is south east of the origin, its position vector will be of the form	+
	<math>k\mathbf{i}-k\mathbf{j}</math>		<math>k\mathbf{i}-k\mathbf{j}</math>
	.		.
-		+
-		+
	<math>\begin{align}		<math>\begin{align}
	& -(9t-180)=5t-450 \\		& -(9t-180)=5t-450 \\

Ian den 2 september 2009 kl. 08.07

2009-09-02T08:07:28Z

← Äldre version		Versionen från 2 september 2009 kl. 08.07
Rad 2:		Rad 2:
	{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"		{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
-	{{Selected tab\|[[5. ~~Forces and equilibrium~~\|Theory]]}}	+	{{Selected tab\|[[7. Position Vectors\|Theory]]}}
-	{{Not selected tab\|[[5. Exercises\|Exercises]]}}	+	{{Not selected tab\|[[7. Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|
	\|}		\|}

Ian den 2 september 2009 kl. 07.53

2009-09-02T07:53:38Z

← Äldre version		Versionen från 2 september 2009 kl. 07.53
Rad 1:		Rad 1:
		+	__NOTOC__
		+	{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
		+	\| style="border-bottom:1px solid #797979" width="5px" \|
		+	{{Selected tab\|[[5. Forces and equilibrium\|Theory]]}}
		+	{{Not selected tab\|[[5. Exercises\|Exercises]]}}
		+	\| style="border-bottom:1px solid #797979" width="100%"\|
		+	\|}
		+
		+
	7. Position Vectors		7. Position Vectors

Ian den 24 augusti 2009 kl. 11.34

2009-08-24T11:34:01Z

← Äldre version		Versionen från 24 augusti 2009 kl. 11.34
Rad 33:		Rad 33:
	<math>0\le t\le 4</math>		<math>0\le t\le 4</math>
	.		.
-		+	{\| width="100%" cellspacing="10px" align="center"
-	Time Position of A (rA) Position of B (rB)	+	\|align="left"\| Time
-	0	+	\| valign="top"\|Position of A (<math>{{\mathbf{r}}_{A}}</math>)
-	<math>{{\mathbf{r}}_{A}}=0\mathbf{i}+0\mathbf{j}</math>	+	\| valign="top"\|Position of B (<math>{{\mathbf{r}}_{B}}</math>)
-		+	\|-
-	<math>{{\mathbf{r}}_{B}}=(6\times 0-{{0}^{2}})\mathbf{i}+0\mathbf{j}=0\mathbf{i}+0\mathbf{j}</math>	+	\|0
-		+	\| valign="top"\| <math>{{\mathbf{r}}_{A}}=0\mathbf{i}+0\mathbf{j}</math>
-	1	+	\| valign="top"\|<math>{{\mathbf{r}}_{B}}=(6\times 0-{{0}^{2}})\mathbf{i}+0\mathbf{j}=0\mathbf{i}+0\mathbf{j}</math>
-	<math>{{\mathbf{r}}_{A}}=2\mathbf{i}+1\mathbf{j}</math>	+	\|-
-		+	\| 1
-	<math>{{\mathbf{r}}_{B}}=(6\times 1-{{1}^{2}})\mathbf{i}+1\mathbf{j}=5\mathbf{i}+1\mathbf{j}</math>	+	\|valign="top"\| <math>{{\mathbf{r}}_{A}}=2\mathbf{i}+1\mathbf{j}</math>
-		+	\|valign="top"\| <math>{{\mathbf{r}}_{B}}=(6\times 1-{{1}^{2}})\mathbf{i}+1\mathbf{j}=5\mathbf{i}+1\mathbf{j}</math>
-	2	+	\|-
-	<math>{{\mathbf{r}}_{A}}=4\mathbf{i}+2\mathbf{j}</math>	+	\| 2
-		+	\|valign="top"\|<math>{{\mathbf{r}}_{A}}=4\mathbf{i}+2\mathbf{j}</math>
-	<math>{{\mathbf{r}}_{B}}=(6\times 2-{{2}^{2}})\mathbf{i}+2\mathbf{j}=8\mathbf{i}+2\mathbf{j}</math>	+	\|valign="top"\|<math>{{\mathbf{r}}_{B}}=(6\times 2-{{2}^{2}})\mathbf{i}+2\mathbf{j}=8\mathbf{i}+2\mathbf{j}</math>
-		+	\|-
-	3	+	\|3
-	<math>{{\mathbf{r}}_{A}}=6\mathbf{i}+3\mathbf{j}</math>	+	\|valign="top"\|<math>{{\mathbf{r}}_{A}}=6\mathbf{i}+3\mathbf{j}</math>
-		+	\|valign="top"\|<math>{{\mathbf{r}}_{B}}=(6\times 3-{{3}^{2}})\mathbf{i}+3\mathbf{j}=9\mathbf{i}+3\mathbf{j}</math>
-	<math>{{\mathbf{r}}_{B}}=(6\times 3-{{3}^{2}})\mathbf{i}+3\mathbf{j}=9\mathbf{i}+3\mathbf{j}</math>	+	\|-
-		+	\|4
-	4	+	\|valign="top"\|<math>{{\mathbf{r}}_{A}}=8\mathbf{i}+4\mathbf{j}</math>
-	<math>{{\mathbf{r}}_{A}}=8\mathbf{i}+4\mathbf{j}</math>	+	\|valign="top"\|<math>{{\mathbf{r}}_{B}}=(6\times 4-{{4}^{2}})\mathbf{i}+4\mathbf{j}=8\mathbf{i}+4\mathbf{j}</math>
-		+	\|}
-	<math>{{\mathbf{r}}_{B}}=(6\times 4-{{4}^{2}})\mathbf{i}+4\mathbf{j}=8\mathbf{i}+4\mathbf{j}</math>	+

Ian den 24 augusti 2009 kl. 10.30

2009-08-24T10:30:04Z

← Äldre version		Versionen från 24 augusti 2009 kl. 10.30
Rad 65:		Rad 65:
	and unless they take evasive action will collide. The paths are shown in the diagram below.		and unless they take evasive action will collide. The paths are shown in the diagram below.

-		+	[[Image:TF7.1.GIF]]

Ian den 24 augusti 2009 kl. 10.19

2009-08-24T10:19:20Z

← Äldre version		Versionen från 24 augusti 2009 kl. 10.19
Rad 14:		Rad 14:


-	Example 7.1	+	'''[[Example 7.1]]'''
	Two children, A and B, start at the origin and run so that their position vectors in metres at time t seconds are given by:		Two children, A and B, start at the origin and run so that their position vectors in metres at time t seconds are given by:

Rad 29:		Rad 29:
	?		?

-	Solution	+	'''Solution'''
	The tables below show the positions of the children at 1 second intervals for		The tables below show the positions of the children at 1 second intervals for
	<math>0\le t\le 4</math>		<math>0\le t\le 4</math>
Rad 67:		Rad 67:


-	Example 7.2	+
		+	'''[[Example 7.2]]'''
	A boat moves so that at time t its position vector is r, where		A boat moves so that at time t its position vector is r, where

Rad 78:		Rad 79:
	(c) Find the position of the boat when it is south east of the origin.		(c) Find the position of the boat when it is south east of the origin.

-	Solution	+	'''Solution'''
	(a) When the boat is due east of the origin the position vector will contain only		(a) When the boat is due east of the origin the position vector will contain only
	<math>\mathbf{i}</math>		<math>\mathbf{i}</math>

Ian: New page: 7. Position Vectors Key Points Velocities can be expressed as vectors, for example $\mathbf{v}=V\cos \alpha \mathbf{i}+V\sin \alpha \mathbf{j}$ When moving with a constan...

2009-08-19T13:01:27Z

New page: 7. Position Vectors Key Points Velocities can be expressed as vectors, for example <math>\mathbf{v}=V\cos \alpha \mathbf{i}+V\sin \alpha \mathbf{j}</math> When moving with a constan...

Ny sida

7. Position Vectors

Key Points

Velocities can be expressed as vectors, for example

<math>\mathbf{v}=V\cos \alpha \mathbf{i}+V\sin \alpha \mathbf{j}</math>

When moving with a constant velocity

<math>\mathbf{r}=\mathbf{u}t+{{\mathbf{r}}_{0}}</math>

Example 7.1
Two children, A and B, start at the origin and run so that their position vectors in metres at time t seconds are given by:

<math>{{\mathbf{r}}_{A}}=2t\mathbf{i}+t\mathbf{j}</math>
and
<math>{{\mathbf{r}}_{B}}=(6t-{{t}^{2}})\mathbf{i}+t\mathbf{j}</math>

Plot the paths of the two children for
<math>0\le t\le 4</math>
. What happens when
<math>t=4</math>
?

Solution
The tables below show the positions of the children at 1 second intervals for
<math>0\le t\le 4</math>
.

Time Position of A (rA) Position of B (rB)
0
<math>{{\mathbf{r}}_{A}}=0\mathbf{i}+0\mathbf{j}</math>

<math>{{\mathbf{r}}_{B}}=(6\times 0-{{0}^{2}})\mathbf{i}+0\mathbf{j}=0\mathbf{i}+0\mathbf{j}</math>

1
<math>{{\mathbf{r}}_{A}}=2\mathbf{i}+1\mathbf{j}</math>

<math>{{\mathbf{r}}_{B}}=(6\times 1-{{1}^{2}})\mathbf{i}+1\mathbf{j}=5\mathbf{i}+1\mathbf{j}</math>

2
<math>{{\mathbf{r}}_{A}}=4\mathbf{i}+2\mathbf{j}</math>

<math>{{\mathbf{r}}_{B}}=(6\times 2-{{2}^{2}})\mathbf{i}+2\mathbf{j}=8\mathbf{i}+2\mathbf{j}</math>

3
<math>{{\mathbf{r}}_{A}}=6\mathbf{i}+3\mathbf{j}</math>

<math>{{\mathbf{r}}_{B}}=(6\times 3-{{3}^{2}})\mathbf{i}+3\mathbf{j}=9\mathbf{i}+3\mathbf{j}</math>

4
<math>{{\mathbf{r}}_{A}}=8\mathbf{i}+4\mathbf{j}</math>

<math>{{\mathbf{r}}_{B}}=(6\times 4-{{4}^{2}})\mathbf{i}+4\mathbf{j}=8\mathbf{i}+4\mathbf{j}</math>

From the values that we have obtained it is clear that the two children will have the same position when
<math>t=4</math>
and unless they take evasive action will collide. The paths are shown in the diagram below.

Example 7.2
A boat moves so that at time t its position vector is r, where

<math>\mathbf{r}=(9t-180)\mathbf{i}+(5t-450)\mathbf{j}</math>

and i and j are unit vectors directed due east and north respectively.

(a) Find the time when the boat is due east of the origin.
(b) Find the time when it is due south of the origin.
(c) Find the position of the boat when it is south east of the origin.

Solution
(a) When the boat is due east of the origin the position vector will contain only
<math>\mathbf{i}</math>
terms and no
<math>\mathbf{j}</math>
terms.

<math>\begin{align}
& 5t-450=0 \\
& t=\frac{450}{5}=90\text{ s} \\
\end{align}</math>

(b) When the boat is due south of the origin the position vector will contain only
<math>\mathbf{j}</math>
terms and no
<math>\mathbf{i}</math>
terms.

<math>\begin{align}
& 9t-180=0 \\
& t=\frac{180}{9}=20\text{ s} \\
\end{align}</math>

(c) When the boat is south east of the origin, its position vector will be of the form
<math>k\mathbf{i}-k\mathbf{j}</math>
.

<math>\begin{align}
& -(9t-180)=5t-450 \\
& -9t+180=5t-450 \\
& 630=14t \\
& t=\frac{630}{14}=45\text{ s} \\
\end{align}</math>

← Äldre version		Versionen från 2 september 2009 kl. 08.07
Rad 2:		Rad 2:
	{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"		{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
-	{{Selected tab\|[[5. ~~Forces and equilibrium~~\|Theory]]}}	+	{{Selected tab\|[[7. Position Vectors\|Theory]]}}
-	{{Not selected tab\|[[5. Exercises\|Exercises]]}}	+	{{Not selected tab\|[[7. Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|
	\|}		\|}