How do I find the volume of a parallelepiped given 4 vertices?
$begingroup$
"Find the volume of the parallelepiped by four vertices: $(0,1,0), (2,2,2), (0,3,0),$ and $(3,1,2)$.
I know the formula to find this volume is: $|vec{a} circ(vec{b}times vec{c})|$, and I know how to carry out the computation to get the actual value. What I need to know is the process of how I set up the values of the vectors $vec{a},vec{b},$ and $vec{c}$ using the given points?
calculus multivariable-calculus volume
edited Dec 16 '18 at 0:32
Key Flex
8,28261233
asked Dec 16 '18 at 0:26
JaigusJaigus
2259
$endgroup$
add a comment |
$begingroup$
"Find the volume of the parallelepiped by four vertices: $(0,1,0), (2,2,2), (0,3,0),$ and $(3,1,2)$.
I know the formula to find this volume is: $|vec{a} circ(vec{b}times vec{c})|$, and I know how to carry out the computation to get the actual value. What I need to know is the process of how I set up the values of the vectors $vec{a},vec{b},$ and $vec{c}$ using the given points?
calculus multivariable-calculus volume
edited Dec 16 '18 at 0:32
Key Flex
8,28261233
asked Dec 16 '18 at 0:26
JaigusJaigus
2259
$endgroup$
$begingroup$
Form three difference vectors at a vertex and find scalar triple product ...
$endgroup$
– Narasimham
Dec 16 '18 at 7:09
add a comment |
$begingroup$
"Find the volume of the parallelepiped by four vertices: $(0,1,0), (2,2,2), (0,3,0),$ and $(3,1,2)$.
I know the formula to find this volume is: $|vec{a} circ(vec{b}times vec{c})|$, and I know how to carry out the computation to get the actual value. What I need to know is the process of how I set up the values of the vectors $vec{a},vec{b},$ and $vec{c}$ using the given points?
calculus multivariable-calculus volume
edited Dec 16 '18 at 0:32
Key Flex
8,28261233
asked Dec 16 '18 at 0:26
JaigusJaigus
2259
$endgroup$
"Find the volume of the parallelepiped by four vertices: $(0,1,0), (2,2,2), (0,3,0),$ and $(3,1,2)$.
I know the formula to find this volume is: $|vec{a} circ(vec{b}times vec{c})|$, and I know how to carry out the computation to get the actual value. What I need to know is the process of how I set up the values of the vectors $vec{a},vec{b},$ and $vec{c}$ using the given points?
calculus multivariable-calculus volume
calculus multivariable-calculus volume
edited Dec 16 '18 at 0:32
Key Flex
8,28261233
asked Dec 16 '18 at 0:26
JaigusJaigus
2259
edited Dec 16 '18 at 0:32
Key Flex
8,28261233
asked Dec 16 '18 at 0:26
JaigusJaigus
2259
edited Dec 16 '18 at 0:32
Key Flex
8,28261233
edited Dec 16 '18 at 0:32
Key Flex
8,28261233
edited Dec 16 '18 at 0:32
Key Flex
8,28261233
8,28261233
asked Dec 16 '18 at 0:26
JaigusJaigus
2259
asked Dec 16 '18 at 0:26
JaigusJaigus
2259
asked Dec 16 '18 at 0:26
JaigusJaigus
2259
2259
$begingroup$
Form three difference vectors at a vertex and find scalar triple product ...
$endgroup$
– Narasimham
Dec 16 '18 at 7:09
add a comment |
$begingroup$
Form three difference vectors at a vertex and find scalar triple product ...
$endgroup$
– Narasimham
Dec 16 '18 at 7:09
$begingroup$
Form three difference vectors at a vertex and find scalar triple product ...
$endgroup$
– Narasimham
Dec 16 '18 at 7:09
$begingroup$
Form three difference vectors at a vertex and find scalar triple product ...
$endgroup$
– Narasimham
Dec 16 '18 at 7:09
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The volume of a parallelepiped determined by the vectors a, b ,c (where a, b and c share the same initial point) is the magnitude of their scalar triple product:
Take four points as $P=(0,1,0),Q=(2,2,2),R=(0,3,0),S=(3,1,2)$ and find
$$PQ=a=langle2-0,2-1,2-0rangle=langle2,1,2rangle$$
$$PR=b=langle0-0,3-1,0-0rangle=langle0,2,0rangle$$
$$PS=c=langle3-0,1-1,2-0rangle=langle3,0,2rangle$$
Then find $|acdot(btimes c)|$
answered Dec 16 '18 at 0:28
Key FlexKey Flex
8,28261233
$endgroup$
$begingroup$
Thanks, this is great, but can you just elaborate a bit about PQ, PR, PS, i.e. why you chose them the way you did, what they represent, and would it matter if we arranged them in a different pairing (such as QR, QS, etc.).
$endgroup$
– Jaigus
Dec 16 '18 at 0:41
$begingroup$
@Jaigus Yes, you can rearrange in any manner like $QR,QS$. Since the volume is determined by vectors, we first find the distance between any two points to get the vectors $a,b,c$.
$endgroup$
– Key Flex
Dec 16 '18 at 1:10
add a comment |
$begingroup$
Translate the parallelepiped such that one of the vertices is the origin. Then the volume has not changed and you can use your formula.
answered Dec 16 '18 at 0:28
SmileyCraftSmileyCraft
3,601517
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The volume of a parallelepiped determined by the vectors a, b ,c (where a, b and c share the same initial point) is the magnitude of their scalar triple product:
Take four points as $P=(0,1,0),Q=(2,2,2),R=(0,3,0),S=(3,1,2)$ and find
$$PQ=a=langle2-0,2-1,2-0rangle=langle2,1,2rangle$$
$$PR=b=langle0-0,3-1,0-0rangle=langle0,2,0rangle$$
$$PS=c=langle3-0,1-1,2-0rangle=langle3,0,2rangle$$
Then find $|acdot(btimes c)|$
answered Dec 16 '18 at 0:28
Key FlexKey Flex
8,28261233
$endgroup$
$begingroup$
Thanks, this is great, but can you just elaborate a bit about PQ, PR, PS, i.e. why you chose them the way you did, what they represent, and would it matter if we arranged them in a different pairing (such as QR, QS, etc.).
$endgroup$
– Jaigus
Dec 16 '18 at 0:41
$begingroup$
@Jaigus Yes, you can rearrange in any manner like $QR,QS$. Since the volume is determined by vectors, we first find the distance between any two points to get the vectors $a,b,c$.
$endgroup$
– Key Flex
Dec 16 '18 at 1:10
add a comment |
$begingroup$
The volume of a parallelepiped determined by the vectors a, b ,c (where a, b and c share the same initial point) is the magnitude of their scalar triple product:
Take four points as $P=(0,1,0),Q=(2,2,2),R=(0,3,0),S=(3,1,2)$ and find
$$PQ=a=langle2-0,2-1,2-0rangle=langle2,1,2rangle$$
$$PR=b=langle0-0,3-1,0-0rangle=langle0,2,0rangle$$
$$PS=c=langle3-0,1-1,2-0rangle=langle3,0,2rangle$$
Then find $|acdot(btimes c)|$
answered Dec 16 '18 at 0:28
Key FlexKey Flex
8,28261233
$endgroup$
$begingroup$
Thanks, this is great, but can you just elaborate a bit about PQ, PR, PS, i.e. why you chose them the way you did, what they represent, and would it matter if we arranged them in a different pairing (such as QR, QS, etc.).
$endgroup$
– Jaigus
Dec 16 '18 at 0:41
$begingroup$
@Jaigus Yes, you can rearrange in any manner like $QR,QS$. Since the volume is determined by vectors, we first find the distance between any two points to get the vectors $a,b,c$.
$endgroup$
– Key Flex
Dec 16 '18 at 1:10
add a comment |
$begingroup$
The volume of a parallelepiped determined by the vectors a, b ,c (where a, b and c share the same initial point) is the magnitude of their scalar triple product:
Take four points as $P=(0,1,0),Q=(2,2,2),R=(0,3,0),S=(3,1,2)$ and find
$$PQ=a=langle2-0,2-1,2-0rangle=langle2,1,2rangle$$
$$PR=b=langle0-0,3-1,0-0rangle=langle0,2,0rangle$$
$$PS=c=langle3-0,1-1,2-0rangle=langle3,0,2rangle$$
Then find $|acdot(btimes c)|$
answered Dec 16 '18 at 0:28
Key FlexKey Flex
8,28261233
$endgroup$
The volume of a parallelepiped determined by the vectors a, b ,c (where a, b and c share the same initial point) is the magnitude of their scalar triple product:
Take four points as $P=(0,1,0),Q=(2,2,2),R=(0,3,0),S=(3,1,2)$ and find
$$PQ=a=langle2-0,2-1,2-0rangle=langle2,1,2rangle$$
$$PR=b=langle0-0,3-1,0-0rangle=langle0,2,0rangle$$
$$PS=c=langle3-0,1-1,2-0rangle=langle3,0,2rangle$$
Then find $|acdot(btimes c)|$
answered Dec 16 '18 at 0:28
Key FlexKey Flex
8,28261233
answered Dec 16 '18 at 0:28
Key FlexKey Flex
8,28261233
answered Dec 16 '18 at 0:28
Key FlexKey Flex
8,28261233
answered Dec 16 '18 at 0:28
Key FlexKey Flex
8,28261233
8,28261233
$begingroup$
Thanks, this is great, but can you just elaborate a bit about PQ, PR, PS, i.e. why you chose them the way you did, what they represent, and would it matter if we arranged them in a different pairing (such as QR, QS, etc.).
$endgroup$
– Jaigus
Dec 16 '18 at 0:41
$begingroup$
@Jaigus Yes, you can rearrange in any manner like $QR,QS$. Since the volume is determined by vectors, we first find the distance between any two points to get the vectors $a,b,c$.
$endgroup$
– Key Flex
Dec 16 '18 at 1:10
add a comment |
$begingroup$
Thanks, this is great, but can you just elaborate a bit about PQ, PR, PS, i.e. why you chose them the way you did, what they represent, and would it matter if we arranged them in a different pairing (such as QR, QS, etc.).
$endgroup$
– Jaigus
Dec 16 '18 at 0:41
$begingroup$
@Jaigus Yes, you can rearrange in any manner like $QR,QS$. Since the volume is determined by vectors, we first find the distance between any two points to get the vectors $a,b,c$.
$endgroup$
– Key Flex
Dec 16 '18 at 1:10
$begingroup$
Thanks, this is great, but can you just elaborate a bit about PQ, PR, PS, i.e. why you chose them the way you did, what they represent, and would it matter if we arranged them in a different pairing (such as QR, QS, etc.).
$endgroup$
– Jaigus
Dec 16 '18 at 0:41
$begingroup$
Thanks, this is great, but can you just elaborate a bit about PQ, PR, PS, i.e. why you chose them the way you did, what they represent, and would it matter if we arranged them in a different pairing (such as QR, QS, etc.).
$endgroup$
– Jaigus
Dec 16 '18 at 0:41
$begingroup$
@Jaigus Yes, you can rearrange in any manner like $QR,QS$. Since the volume is determined by vectors, we first find the distance between any two points to get the vectors $a,b,c$.
$endgroup$
– Key Flex
Dec 16 '18 at 1:10
$begingroup$
@Jaigus Yes, you can rearrange in any manner like $QR,QS$. Since the volume is determined by vectors, we first find the distance between any two points to get the vectors $a,b,c$.
$endgroup$
– Key Flex
Dec 16 '18 at 1:10
add a comment |
$begingroup$
Translate the parallelepiped such that one of the vertices is the origin. Then the volume has not changed and you can use your formula.
answered Dec 16 '18 at 0:28
SmileyCraftSmileyCraft
3,601517
$endgroup$
add a comment |
$begingroup$
Translate the parallelepiped such that one of the vertices is the origin. Then the volume has not changed and you can use your formula.
answered Dec 16 '18 at 0:28
SmileyCraftSmileyCraft
3,601517
$endgroup$
add a comment |
$begingroup$
Translate the parallelepiped such that one of the vertices is the origin. Then the volume has not changed and you can use your formula.
answered Dec 16 '18 at 0:28
SmileyCraftSmileyCraft
3,601517
$endgroup$
Translate the parallelepiped such that one of the vertices is the origin. Then the volume has not changed and you can use your formula.
answered Dec 16 '18 at 0:28
SmileyCraftSmileyCraft
3,601517
answered Dec 16 '18 at 0:28
SmileyCraftSmileyCraft
3,601517
answered Dec 16 '18 at 0:28
SmileyCraftSmileyCraft
3,601517
answered Dec 16 '18 at 0:28
SmileyCraftSmileyCraft
3,601517
3,601517
add a comment |
add a comment |
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$begingroup$
Form three difference vectors at a vertex and find scalar triple product ...
$endgroup$
– Narasimham
Dec 16 '18 at 7:09