Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff...
$begingroup$
This question already has an answer here:
Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$
1 answer
A.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff
$bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1=r$ which $r$ is a real number.
B.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $lambda_1,lambda_2,lambda_3$ such that $lambda_1+lambda_2+lambda_3=0$ and $lambda_1z_1+lambda_2z_2+lambda_3z_3=0$
I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.
complex-analysis complex-numbers
edited Jan 11 at 15:14
David C. Ullrich
60k43994
asked Jan 11 at 15:12
Jaqen ChouJaqen Chou
422110
$endgroup$
marked as duplicate by Martin R, A. Pongrácz, Robert Soupe, amWhy, José Carlos Santos complex-analysis
Users with the complex-analysis badge can single-handedly close complex-analysis questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Jan 14 at 20:36
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$
1 answer
A.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff
$bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1=r$ which $r$ is a real number.
B.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $lambda_1,lambda_2,lambda_3$ such that $lambda_1+lambda_2+lambda_3=0$ and $lambda_1z_1+lambda_2z_2+lambda_3z_3=0$
I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.
complex-analysis complex-numbers
edited Jan 11 at 15:14
David C. Ullrich
60k43994
asked Jan 11 at 15:12
Jaqen ChouJaqen Chou
422110
$endgroup$
marked as duplicate by Martin R, A. Pongrácz, Robert Soupe, amWhy, José Carlos Santos complex-analysis
Users with the complex-analysis badge can single-handedly close complex-analysis questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Jan 14 at 20:36
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
You could start deriving B. from the assertion you already know?
$endgroup$
– Matteo
Jan 11 at 15:20
1
$begingroup$
I just found that I could prove B.
$endgroup$
– Jaqen Chou
Jan 11 at 15:46
$begingroup$
Be careful that your problem requires checking both necessary and sufficient condition.
$endgroup$
– Matteo
Jan 11 at 16:14
$begingroup$
Duplicate of math.stackexchange.com/q/921220
$endgroup$
– Jean Marie
Jan 11 at 16:42
add a comment |
$begingroup$
This question already has an answer here:
Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$
1 answer
A.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff
$bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1=r$ which $r$ is a real number.
B.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $lambda_1,lambda_2,lambda_3$ such that $lambda_1+lambda_2+lambda_3=0$ and $lambda_1z_1+lambda_2z_2+lambda_3z_3=0$
I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.
complex-analysis complex-numbers
edited Jan 11 at 15:14
David C. Ullrich
60k43994
asked Jan 11 at 15:12
Jaqen ChouJaqen Chou
422110
$endgroup$
This question already has an answer here:
Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$
1 answer
A.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff
$bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1=r$ which $r$ is a real number.
B.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $lambda_1,lambda_2,lambda_3$ such that $lambda_1+lambda_2+lambda_3=0$ and $lambda_1z_1+lambda_2z_2+lambda_3z_3=0$
I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.
This question already has an answer here:
Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$
1 answer
complex-analysis complex-numbers
complex-analysis complex-numbers
edited Jan 11 at 15:14
David C. Ullrich
60k43994
asked Jan 11 at 15:12
Jaqen ChouJaqen Chou
422110
edited Jan 11 at 15:14
David C. Ullrich
60k43994
asked Jan 11 at 15:12
Jaqen ChouJaqen Chou
422110
edited Jan 11 at 15:14
David C. Ullrich
60k43994
edited Jan 11 at 15:14
David C. Ullrich
60k43994
edited Jan 11 at 15:14
David C. Ullrich
60k43994
60k43994
asked Jan 11 at 15:12
Jaqen ChouJaqen Chou
422110
asked Jan 11 at 15:12
Jaqen ChouJaqen Chou
422110
asked Jan 11 at 15:12
Jaqen ChouJaqen Chou
422110
422110
marked as duplicate by Martin R, A. Pongrácz, Robert Soupe, amWhy, José Carlos Santos complex-analysis
Users with the complex-analysis badge can single-handedly close complex-analysis questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Jan 14 at 20:36
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Martin R, A. Pongrácz, Robert Soupe, amWhy, José Carlos Santos complex-analysis
Users with the complex-analysis badge can single-handedly close complex-analysis questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Jan 14 at 20:36
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
You could start deriving B. from the assertion you already know?
$endgroup$
– Matteo
Jan 11 at 15:20
1
$begingroup$
I just found that I could prove B.
$endgroup$
– Jaqen Chou
Jan 11 at 15:46
$begingroup$
Be careful that your problem requires checking both necessary and sufficient condition.
$endgroup$
– Matteo
Jan 11 at 16:14
$begingroup$
Duplicate of math.stackexchange.com/q/921220
$endgroup$
– Jean Marie
Jan 11 at 16:42
add a comment |
$begingroup$
You could start deriving B. from the assertion you already know?
$endgroup$
– Matteo
Jan 11 at 15:20
1
$begingroup$
I just found that I could prove B.
$endgroup$
– Jaqen Chou
Jan 11 at 15:46
$begingroup$
Be careful that your problem requires checking both necessary and sufficient condition.
$endgroup$
– Matteo
Jan 11 at 16:14
$begingroup$
Duplicate of math.stackexchange.com/q/921220
$endgroup$
– Jean Marie
Jan 11 at 16:42
$begingroup$
You could start deriving B. from the assertion you already know?
$endgroup$
– Matteo
Jan 11 at 15:20
$begingroup$
You could start deriving B. from the assertion you already know?
$endgroup$
– Matteo
Jan 11 at 15:20
1
1
$begingroup$
I just found that I could prove B.
$endgroup$
– Jaqen Chou
Jan 11 at 15:46
$begingroup$
I just found that I could prove B.
$endgroup$
– Jaqen Chou
Jan 11 at 15:46
$begingroup$
Be careful that your problem requires checking both necessary and sufficient condition.
$endgroup$
– Matteo
Jan 11 at 16:14
$begingroup$
Be careful that your problem requires checking both necessary and sufficient condition.
$endgroup$
– Matteo
Jan 11 at 16:14
$begingroup$
Duplicate of math.stackexchange.com/q/921220
$endgroup$
– Jean Marie
Jan 11 at 16:42
$begingroup$
Duplicate of math.stackexchange.com/q/921220
$endgroup$
– Jean Marie
Jan 11 at 16:42
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :
Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is
$$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$
(see for example Equation of line in form of determinant)
Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :
$$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$
which, once developed, is equivalent to
$$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$
But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.
answered Jan 11 at 17:10
Jean MarieJean Marie
29.2k42050
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :
Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is
$$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$
(see for example Equation of line in form of determinant)
Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :
$$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$
which, once developed, is equivalent to
$$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$
But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.
answered Jan 11 at 17:10
Jean MarieJean Marie
29.2k42050
$endgroup$
add a comment |
$begingroup$
I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :
Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is
$$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$
(see for example Equation of line in form of determinant)
Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :
$$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$
which, once developed, is equivalent to
$$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$
But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.
answered Jan 11 at 17:10
Jean MarieJean Marie
29.2k42050
$endgroup$
add a comment |
$begingroup$
I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :
Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is
$$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$
(see for example Equation of line in form of determinant)
Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :
$$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$
which, once developed, is equivalent to
$$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$
But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.
answered Jan 11 at 17:10
Jean MarieJean Marie
29.2k42050
$endgroup$
I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :
Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is
$$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$
(see for example Equation of line in form of determinant)
Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :
$$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$
which, once developed, is equivalent to
$$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$
But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.
answered Jan 11 at 17:10
Jean MarieJean Marie
29.2k42050
edited Jan 11 at 23:18
answered Jan 11 at 17:10
Jean MarieJean Marie
29.2k42050
answered Jan 11 at 17:10
Jean MarieJean Marie
29.2k42050
answered Jan 11 at 17:10
Jean MarieJean Marie
29.2k42050
29.2k42050
add a comment |
add a comment |
$begingroup$
You could start deriving B. from the assertion you already know?
$endgroup$
– Matteo
Jan 11 at 15:20
1
$begingroup$
I just found that I could prove B.
$endgroup$
– Jaqen Chou
Jan 11 at 15:46
$begingroup$
Be careful that your problem requires checking both necessary and sufficient condition.
$endgroup$
– Matteo
Jan 11 at 16:14
$begingroup$
Duplicate of math.stackexchange.com/q/921220
$endgroup$
– Jean Marie
Jan 11 at 16:42