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Deduce a result about parallelograms
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I have a problem with an exercise from David Poole's Linear Algebra: A Modern Introduction.
(a) Prove that $||mathbf u + mathbf v||^2 + ||mathbf u - mathbf v||^2 = 2||mathbf u||^2 + 2||mathbf v||^2$ for all vectors $mathbf u$ and $mathbf v$ in $mathbb R^n$.
(b) Draw a diagram showing $mathbf u$, $mathbf v$, $mathbf u + mathbf v$, $mathbf u - mathbf v$ in $mathbb R^2$ and use (a) to deduce a result about parallelograms.
I did (a) and drew some diagrams, but I can't connect the dots and I don't know what I'm supposed to deduce.
linear-algebra
asked Jul 25 at 12:30
Dumb Dumb
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up vote
2
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I have a problem with an exercise from David Poole's Linear Algebra: A Modern Introduction.
(a) Prove that $||mathbf u + mathbf v||^2 + ||mathbf u - mathbf v||^2 = 2||mathbf u||^2 + 2||mathbf v||^2$ for all vectors $mathbf u$ and $mathbf v$ in $mathbb R^n$.
(b) Draw a diagram showing $mathbf u$, $mathbf v$, $mathbf u + mathbf v$, $mathbf u - mathbf v$ in $mathbb R^2$ and use (a) to deduce a result about parallelograms.
I did (a) and drew some diagrams, but I can't connect the dots and I don't know what I'm supposed to deduce.
linear-algebra
asked Jul 25 at 12:30
Dumb Dumb
293
It's very hard to imagine where you could be stuck here, you should include a draft of your diagrams in one way or another. Although, a hint: instead of dots as you mentioned, use arrows to represent vectors.
– Arnaud Mortier
Jul 25 at 12:42
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up vote
2
down vote
favorite
up vote
2
down vote
favorite
I have a problem with an exercise from David Poole's Linear Algebra: A Modern Introduction.
(a) Prove that $||mathbf u + mathbf v||^2 + ||mathbf u - mathbf v||^2 = 2||mathbf u||^2 + 2||mathbf v||^2$ for all vectors $mathbf u$ and $mathbf v$ in $mathbb R^n$.
(b) Draw a diagram showing $mathbf u$, $mathbf v$, $mathbf u + mathbf v$, $mathbf u - mathbf v$ in $mathbb R^2$ and use (a) to deduce a result about parallelograms.
I did (a) and drew some diagrams, but I can't connect the dots and I don't know what I'm supposed to deduce.
linear-algebra
asked Jul 25 at 12:30
Dumb Dumb
293
I have a problem with an exercise from David Poole's Linear Algebra: A Modern Introduction.
(a) Prove that $||mathbf u + mathbf v||^2 + ||mathbf u - mathbf v||^2 = 2||mathbf u||^2 + 2||mathbf v||^2$ for all vectors $mathbf u$ and $mathbf v$ in $mathbb R^n$.
(b) Draw a diagram showing $mathbf u$, $mathbf v$, $mathbf u + mathbf v$, $mathbf u - mathbf v$ in $mathbb R^2$ and use (a) to deduce a result about parallelograms.
I did (a) and drew some diagrams, but I can't connect the dots and I don't know what I'm supposed to deduce.
linear-algebra
asked Jul 25 at 12:30
Dumb Dumb
293
asked Jul 25 at 12:30
Dumb Dumb
293
asked Jul 25 at 12:30
Dumb Dumb
293
asked Jul 25 at 12:30
Dumb Dumb
293
293
It's very hard to imagine where you could be stuck here, you should include a draft of your diagrams in one way or another. Although, a hint: instead of dots as you mentioned, use arrows to represent vectors.
– Arnaud Mortier
Jul 25 at 12:42
add a comment |Â
It's very hard to imagine where you could be stuck here, you should include a draft of your diagrams in one way or another. Although, a hint: instead of dots as you mentioned, use arrows to represent vectors.
– Arnaud Mortier
Jul 25 at 12:42
It's very hard to imagine where you could be stuck here, you should include a draft of your diagrams in one way or another. Although, a hint: instead of dots as you mentioned, use arrows to represent vectors.
– Arnaud Mortier
Jul 25 at 12:42
It's very hard to imagine where you could be stuck here, you should include a draft of your diagrams in one way or another. Although, a hint: instead of dots as you mentioned, use arrows to represent vectors.
– Arnaud Mortier
Jul 25 at 12:42
add a comment |Â
2 Answers
2
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1
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Think of the parallelogram having four vertices $0, u, v$, and $u+v$. The lengths of the two diagonals of the parallelogram are $| u + v |$ and $| u - v |$. The perimeter of the parallelogram is $2 | u | + 2 | v |$.
You've shown that the sum of squares of the two diagonals is equal to the sum of the squares of the sides making the perimeter. This is called the parallelogram law.
answered Jul 25 at 12:43
davidlowryduda♦
72.5k6110242
I see now. Thank you very much.
– Dumb Dumb
Jul 25 at 12:47
1
@DumbDumb Welcome to Math.SE! As you are a new user and apparently you found this answer helpful, I thought I should describe the voting system here. It's good to upvote posts that helped you or that you like, as this helps direct future users with similar questions to good content. As an asker, you also have the ability to mark an answer as "accepted", indicating that this is the answer to look at if others have the same problem, and that the problem is resolved. Downvoting is similarly important. The site functions because good content is identifiable through voting. Vote early, vote often!
– davidlowryduda♦
Jul 25 at 12:54
add a comment |Â
up vote
1
down vote
I disagree with the answer by davidlowryduda. He is not taking into account the exponent 2 that appears in the equation. (Edit: His answer has now been corrected.)
Consider the parallelogram he describes. I agree with him that the diagonals have lengths $||u + v||$ and $||u - v||$, and that the side lengths are $||u||$, $||v||$, $||u||$, $||v||$. You should be able to formulate the result as a statement about the lengths of the sides and diagonals of a parallelogram.
However, logically, you should not start with $u$ and $v$ and draw the parallelogram. Instead, you should consider an arbitrary parallelogram $ABCD$, give the names $u$ and $v$ to the vectors $AB$ and $AD$, and then invoke the identity you proved. That way the argument will be applicable to any parallelogram.
answered Jul 25 at 12:53
Dave
1262
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Think of the parallelogram having four vertices $0, u, v$, and $u+v$. The lengths of the two diagonals of the parallelogram are $| u + v |$ and $| u - v |$. The perimeter of the parallelogram is $2 | u | + 2 | v |$.
You've shown that the sum of squares of the two diagonals is equal to the sum of the squares of the sides making the perimeter. This is called the parallelogram law.
answered Jul 25 at 12:43
davidlowryduda♦
72.5k6110242
I see now. Thank you very much.
– Dumb Dumb
Jul 25 at 12:47
1
@DumbDumb Welcome to Math.SE! As you are a new user and apparently you found this answer helpful, I thought I should describe the voting system here. It's good to upvote posts that helped you or that you like, as this helps direct future users with similar questions to good content. As an asker, you also have the ability to mark an answer as "accepted", indicating that this is the answer to look at if others have the same problem, and that the problem is resolved. Downvoting is similarly important. The site functions because good content is identifiable through voting. Vote early, vote often!
– davidlowryduda♦
Jul 25 at 12:54
add a comment |Â
up vote
1
down vote
accepted
Think of the parallelogram having four vertices $0, u, v$, and $u+v$. The lengths of the two diagonals of the parallelogram are $| u + v |$ and $| u - v |$. The perimeter of the parallelogram is $2 | u | + 2 | v |$.
You've shown that the sum of squares of the two diagonals is equal to the sum of the squares of the sides making the perimeter. This is called the parallelogram law.
answered Jul 25 at 12:43
davidlowryduda♦
72.5k6110242
I see now. Thank you very much.
– Dumb Dumb
Jul 25 at 12:47
1
@DumbDumb Welcome to Math.SE! As you are a new user and apparently you found this answer helpful, I thought I should describe the voting system here. It's good to upvote posts that helped you or that you like, as this helps direct future users with similar questions to good content. As an asker, you also have the ability to mark an answer as "accepted", indicating that this is the answer to look at if others have the same problem, and that the problem is resolved. Downvoting is similarly important. The site functions because good content is identifiable through voting. Vote early, vote often!
– davidlowryduda♦
Jul 25 at 12:54
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Think of the parallelogram having four vertices $0, u, v$, and $u+v$. The lengths of the two diagonals of the parallelogram are $| u + v |$ and $| u - v |$. The perimeter of the parallelogram is $2 | u | + 2 | v |$.
You've shown that the sum of squares of the two diagonals is equal to the sum of the squares of the sides making the perimeter. This is called the parallelogram law.
answered Jul 25 at 12:43
davidlowryduda♦
72.5k6110242
Think of the parallelogram having four vertices $0, u, v$, and $u+v$. The lengths of the two diagonals of the parallelogram are $| u + v |$ and $| u - v |$. The perimeter of the parallelogram is $2 | u | + 2 | v |$.
You've shown that the sum of squares of the two diagonals is equal to the sum of the squares of the sides making the perimeter. This is called the parallelogram law.
answered Jul 25 at 12:43
davidlowryduda♦
72.5k6110242
edited Jul 25 at 12:56
answered Jul 25 at 12:43
davidlowryduda♦
72.5k6110242
answered Jul 25 at 12:43
davidlowryduda♦
72.5k6110242
answered Jul 25 at 12:43
davidlowryduda♦
72.5k6110242
72.5k6110242
I see now. Thank you very much.
– Dumb Dumb
Jul 25 at 12:47
1
@DumbDumb Welcome to Math.SE! As you are a new user and apparently you found this answer helpful, I thought I should describe the voting system here. It's good to upvote posts that helped you or that you like, as this helps direct future users with similar questions to good content. As an asker, you also have the ability to mark an answer as "accepted", indicating that this is the answer to look at if others have the same problem, and that the problem is resolved. Downvoting is similarly important. The site functions because good content is identifiable through voting. Vote early, vote often!
– davidlowryduda♦
Jul 25 at 12:54
add a comment |Â
I see now. Thank you very much.
– Dumb Dumb
Jul 25 at 12:47
1
@DumbDumb Welcome to Math.SE! As you are a new user and apparently you found this answer helpful, I thought I should describe the voting system here. It's good to upvote posts that helped you or that you like, as this helps direct future users with similar questions to good content. As an asker, you also have the ability to mark an answer as "accepted", indicating that this is the answer to look at if others have the same problem, and that the problem is resolved. Downvoting is similarly important. The site functions because good content is identifiable through voting. Vote early, vote often!
– davidlowryduda♦
Jul 25 at 12:54
I see now. Thank you very much.
– Dumb Dumb
Jul 25 at 12:47
I see now. Thank you very much.
– Dumb Dumb
Jul 25 at 12:47
1
1
@DumbDumb Welcome to Math.SE! As you are a new user and apparently you found this answer helpful, I thought I should describe the voting system here. It's good to upvote posts that helped you or that you like, as this helps direct future users with similar questions to good content. As an asker, you also have the ability to mark an answer as "accepted", indicating that this is the answer to look at if others have the same problem, and that the problem is resolved. Downvoting is similarly important. The site functions because good content is identifiable through voting. Vote early, vote often!
– davidlowryduda♦
Jul 25 at 12:54
@DumbDumb Welcome to Math.SE! As you are a new user and apparently you found this answer helpful, I thought I should describe the voting system here. It's good to upvote posts that helped you or that you like, as this helps direct future users with similar questions to good content. As an asker, you also have the ability to mark an answer as "accepted", indicating that this is the answer to look at if others have the same problem, and that the problem is resolved. Downvoting is similarly important. The site functions because good content is identifiable through voting. Vote early, vote often!
– davidlowryduda♦
Jul 25 at 12:54
add a comment |Â
up vote
1
down vote
I disagree with the answer by davidlowryduda. He is not taking into account the exponent 2 that appears in the equation. (Edit: His answer has now been corrected.)
Consider the parallelogram he describes. I agree with him that the diagonals have lengths $||u + v||$ and $||u - v||$, and that the side lengths are $||u||$, $||v||$, $||u||$, $||v||$. You should be able to formulate the result as a statement about the lengths of the sides and diagonals of a parallelogram.
However, logically, you should not start with $u$ and $v$ and draw the parallelogram. Instead, you should consider an arbitrary parallelogram $ABCD$, give the names $u$ and $v$ to the vectors $AB$ and $AD$, and then invoke the identity you proved. That way the argument will be applicable to any parallelogram.
answered Jul 25 at 12:53
Dave
1262
add a comment |Â
up vote
1
down vote
I disagree with the answer by davidlowryduda. He is not taking into account the exponent 2 that appears in the equation. (Edit: His answer has now been corrected.)
Consider the parallelogram he describes. I agree with him that the diagonals have lengths $||u + v||$ and $||u - v||$, and that the side lengths are $||u||$, $||v||$, $||u||$, $||v||$. You should be able to formulate the result as a statement about the lengths of the sides and diagonals of a parallelogram.
However, logically, you should not start with $u$ and $v$ and draw the parallelogram. Instead, you should consider an arbitrary parallelogram $ABCD$, give the names $u$ and $v$ to the vectors $AB$ and $AD$, and then invoke the identity you proved. That way the argument will be applicable to any parallelogram.
answered Jul 25 at 12:53
Dave
1262
add a comment |Â
up vote
1
down vote
up vote
1
down vote
I disagree with the answer by davidlowryduda. He is not taking into account the exponent 2 that appears in the equation. (Edit: His answer has now been corrected.)
Consider the parallelogram he describes. I agree with him that the diagonals have lengths $||u + v||$ and $||u - v||$, and that the side lengths are $||u||$, $||v||$, $||u||$, $||v||$. You should be able to formulate the result as a statement about the lengths of the sides and diagonals of a parallelogram.
However, logically, you should not start with $u$ and $v$ and draw the parallelogram. Instead, you should consider an arbitrary parallelogram $ABCD$, give the names $u$ and $v$ to the vectors $AB$ and $AD$, and then invoke the identity you proved. That way the argument will be applicable to any parallelogram.
answered Jul 25 at 12:53
Dave
1262
I disagree with the answer by davidlowryduda. He is not taking into account the exponent 2 that appears in the equation. (Edit: His answer has now been corrected.)
Consider the parallelogram he describes. I agree with him that the diagonals have lengths $||u + v||$ and $||u - v||$, and that the side lengths are $||u||$, $||v||$, $||u||$, $||v||$. You should be able to formulate the result as a statement about the lengths of the sides and diagonals of a parallelogram.
However, logically, you should not start with $u$ and $v$ and draw the parallelogram. Instead, you should consider an arbitrary parallelogram $ABCD$, give the names $u$ and $v$ to the vectors $AB$ and $AD$, and then invoke the identity you proved. That way the argument will be applicable to any parallelogram.
answered Jul 25 at 12:53
Dave
1262
edited Jul 25 at 12:59
answered Jul 25 at 12:53
Dave
1262
answered Jul 25 at 12:53
Dave
1262
answered Jul 25 at 12:53
Dave
1262
1262
add a comment |Â
add a comment |Â
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It's very hard to imagine where you could be stuck here, you should include a draft of your diagrams in one way or another. Although, a hint: instead of dots as you mentioned, use arrows to represent vectors.
– Arnaud Mortier
Jul 25 at 12:42