Using trigonometric formulas to prove that $m_1m_2$ = -1 for perpendicular lines? The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Is there a way to solve for $x$ in $cos^-1(ax) / cos^-1(bx) = c$?Angular distribution of lines passing through two squares.Radial velocityIs it true that: $frac 1 10 (sin(y_1+y_2)-sin(x_1+x_2)+y_2-x_2)^2+(cos(x_1+x_2)-cos(y_1+y_2)+x_1-y_1)^2) le (y_1-x_1)^2+(y_2-x_2)^2$?Help with simplifying a trig equationFinding $2$ points from angle.Find the control point of quadratic Bezier curve having only the end-pointsGeometric proof of dot product and projectionSimplifying a parametric equation for the formula of a circle given to point on opposite sidesProving rules for real numbers hold for complex numbers
Who or what is the being for whom Being is a question for Heidegger?
How do you keep chess fun when your opponent constantly beats you?
How many people can fit inside Mordenkainen's Magnificent Mansion?
Is it ok to offer lower paid work as a trial period before negotiating for a full-time job?
Reference for the teaching of not-self
How can I protect witches in combat who wear limited clothing?
I could not break this equation. Please help me
Did the new image of black hole confirm the general theory of relativity?
Why can't wing-mounted spoilers be used to steepen approaches?
does high air pressure throw off wheel balance?
Semisimplicity of the category of coherent sheaves?
Can withdrawing asylum be illegal?
Segmentation fault output is suppressed when piping stdin into a function. Why?
What can I do if neighbor is blocking my solar panels intentionally?
Wolves and sheep
What information about me do stores get via my credit card?
Is there a writing software that you can sort scenes like slides in PowerPoint?
How are presidential pardons supposed to be used?
How to delete random line from file using Unix command?
How did passengers keep warm on sail ships?
Can a novice safely splice in wire to lengthen 5V charging cable?
Can a 1st-level character have an ability score above 18?
What is special about square numbers here?
How to split my screen on my Macbook Air?
Using trigonometric formulas to prove that $m_1m_2$ = -1 for perpendicular lines?
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Is there a way to solve for $x$ in $cos^-1(ax) / cos^-1(bx) = c$?Angular distribution of lines passing through two squares.Radial velocityIs it true that: $frac 1 10 (sin(y_1+y_2)-sin(x_1+x_2)+y_2-x_2)^2+(cos(x_1+x_2)-cos(y_1+y_2)+x_1-y_1)^2) le (y_1-x_1)^2+(y_2-x_2)^2$?Help with simplifying a trig equationFinding $2$ points from angle.Find the control point of quadratic Bezier curve having only the end-pointsGeometric proof of dot product and projectionSimplifying a parametric equation for the formula of a circle given to point on opposite sidesProving rules for real numbers hold for complex numbers
$begingroup$
How do you use trigonometric formulas (or identities) to prove that the product of the gradients of two perpendicular lines is -1?
If
$y = m_1x + c_1 text and y = m_2x + c_2,$
I thought finding an angle would help to incorporate one of the identities, and hence get somewhere. But how do I find an angle? Constructing two vectors in terms of the y-intercepts and x-intercepts of the two equations like below?
($y_1$ is the y-intercept of the first equation, $x_1$ is the x-intercept of the first equation, and so on.)
$
rightarrow left(beginarraycx_1\ y_1endarrayright) text•
left (beginarraycx_2\ y_2endarrayright) \~\
rightarrow left(sqrt(x_1^2 + y_1^2)(x_2^2 + y_2^2)hspace0.08cm right) cos theta = x_1x_2 + y_1y_2 \
theta = cos^-1 left[fracx_1x_2 + y_1y_2sqrt(x_1^2 + y_1^2)(x_2^2 + y_2^2) right]
$
But this is taking me nowhere!
Thanks in advance.
trigonometry alternative-proof
asked Mar 25 at 9:38
RamanaRamana
17210
$endgroup$
add a comment |
$begingroup$
How do you use trigonometric formulas (or identities) to prove that the product of the gradients of two perpendicular lines is -1?
If
$y = m_1x + c_1 text and y = m_2x + c_2,$
I thought finding an angle would help to incorporate one of the identities, and hence get somewhere. But how do I find an angle? Constructing two vectors in terms of the y-intercepts and x-intercepts of the two equations like below?
($y_1$ is the y-intercept of the first equation, $x_1$ is the x-intercept of the first equation, and so on.)
$
rightarrow left(beginarraycx_1\ y_1endarrayright) text•
left (beginarraycx_2\ y_2endarrayright) \~\
rightarrow left(sqrt(x_1^2 + y_1^2)(x_2^2 + y_2^2)hspace0.08cm right) cos theta = x_1x_2 + y_1y_2 \
theta = cos^-1 left[fracx_1x_2 + y_1y_2sqrt(x_1^2 + y_1^2)(x_2^2 + y_2^2) right]
$
But this is taking me nowhere!
Thanks in advance.
trigonometry alternative-proof
asked Mar 25 at 9:38
RamanaRamana
17210
$endgroup$
3
$begingroup$
If you know vectors, you can recall that the lines have direction vectors $beginpmatrix1\ m_1endpmatrix$ and $beginpmatrix1\ m_2endpmatrix$ respectively. Then note that the lines are perpendicular if and only if their directions vectors are perpendicular (i.e. their dot product is $0$). This will get you the result.
$endgroup$
– Minus One-Twelfth
Mar 25 at 9:43
$begingroup$
Minus One-Twelfth.Right your comment as an snsert , I delete mine.OK?
$endgroup$
– Peter Szilas
Mar 25 at 10:04
add a comment |
$begingroup$
How do you use trigonometric formulas (or identities) to prove that the product of the gradients of two perpendicular lines is -1?
If
$y = m_1x + c_1 text and y = m_2x + c_2,$
I thought finding an angle would help to incorporate one of the identities, and hence get somewhere. But how do I find an angle? Constructing two vectors in terms of the y-intercepts and x-intercepts of the two equations like below?
($y_1$ is the y-intercept of the first equation, $x_1$ is the x-intercept of the first equation, and so on.)
$
rightarrow left(beginarraycx_1\ y_1endarrayright) text•
left (beginarraycx_2\ y_2endarrayright) \~\
rightarrow left(sqrt(x_1^2 + y_1^2)(x_2^2 + y_2^2)hspace0.08cm right) cos theta = x_1x_2 + y_1y_2 \
theta = cos^-1 left[fracx_1x_2 + y_1y_2sqrt(x_1^2 + y_1^2)(x_2^2 + y_2^2) right]
$
But this is taking me nowhere!
Thanks in advance.
trigonometry alternative-proof
asked Mar 25 at 9:38
RamanaRamana
17210
$endgroup$
How do you use trigonometric formulas (or identities) to prove that the product of the gradients of two perpendicular lines is -1?
If
$y = m_1x + c_1 text and y = m_2x + c_2,$
I thought finding an angle would help to incorporate one of the identities, and hence get somewhere. But how do I find an angle? Constructing two vectors in terms of the y-intercepts and x-intercepts of the two equations like below?
($y_1$ is the y-intercept of the first equation, $x_1$ is the x-intercept of the first equation, and so on.)
$
rightarrow left(beginarraycx_1\ y_1endarrayright) text•
left (beginarraycx_2\ y_2endarrayright) \~\
rightarrow left(sqrt(x_1^2 + y_1^2)(x_2^2 + y_2^2)hspace0.08cm right) cos theta = x_1x_2 + y_1y_2 \
theta = cos^-1 left[fracx_1x_2 + y_1y_2sqrt(x_1^2 + y_1^2)(x_2^2 + y_2^2) right]
$
But this is taking me nowhere!
Thanks in advance.
trigonometry alternative-proof
trigonometry alternative-proof
asked Mar 25 at 9:38
RamanaRamana
17210
asked Mar 25 at 9:38
RamanaRamana
17210
asked Mar 25 at 9:38
RamanaRamana
17210
asked Mar 25 at 9:38
RamanaRamana
17210
asked Mar 25 at 9:38
RamanaRamana
17210
17210
3
$begingroup$
If you know vectors, you can recall that the lines have direction vectors $beginpmatrix1\ m_1endpmatrix$ and $beginpmatrix1\ m_2endpmatrix$ respectively. Then note that the lines are perpendicular if and only if their directions vectors are perpendicular (i.e. their dot product is $0$). This will get you the result.
$endgroup$
– Minus One-Twelfth
Mar 25 at 9:43
$begingroup$
Minus One-Twelfth.Right your comment as an snsert , I delete mine.OK?
$endgroup$
– Peter Szilas
Mar 25 at 10:04
add a comment |
3
$begingroup$
If you know vectors, you can recall that the lines have direction vectors $beginpmatrix1\ m_1endpmatrix$ and $beginpmatrix1\ m_2endpmatrix$ respectively. Then note that the lines are perpendicular if and only if their directions vectors are perpendicular (i.e. their dot product is $0$). This will get you the result.
$endgroup$
– Minus One-Twelfth
Mar 25 at 9:43
$begingroup$
Minus One-Twelfth.Right your comment as an snsert , I delete mine.OK?
$endgroup$
– Peter Szilas
Mar 25 at 10:04
3
3
$begingroup$
If you know vectors, you can recall that the lines have direction vectors $beginpmatrix1\ m_1endpmatrix$ and $beginpmatrix1\ m_2endpmatrix$ respectively. Then note that the lines are perpendicular if and only if their directions vectors are perpendicular (i.e. their dot product is $0$). This will get you the result.
$endgroup$
– Minus One-Twelfth
Mar 25 at 9:43
$begingroup$
If you know vectors, you can recall that the lines have direction vectors $beginpmatrix1\ m_1endpmatrix$ and $beginpmatrix1\ m_2endpmatrix$ respectively. Then note that the lines are perpendicular if and only if their directions vectors are perpendicular (i.e. their dot product is $0$). This will get you the result.
$endgroup$
– Minus One-Twelfth
Mar 25 at 9:43
$begingroup$
Minus One-Twelfth.Right your comment as an snsert , I delete mine.OK?
$endgroup$
– Peter Szilas
Mar 25 at 10:04
$begingroup$
Minus One-Twelfth.Right your comment as an snsert , I delete mine.OK?
$endgroup$
– Peter Szilas
Mar 25 at 10:04
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If $theta$ is the angle made by the first line with $x-$ axis then the slope of this line is $tan (theta)$. The slope of the second line is $tan (pi /2+theta)=-cot(theta)$. Since $tan (theta)$ $(-cot (theta))=-1$ we are done.
answered Mar 25 at 9:41
Kavi Rama MurthyKavi Rama Murthy
74.6k53270
$endgroup$
$begingroup$
Thanks a lot, sir.
$endgroup$
– Ramana
Mar 25 at 9:43
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3161568%2fusing-trigonometric-formulas-to-prove-that-m-1m-2-1-for-perpendicular-lines%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If $theta$ is the angle made by the first line with $x-$ axis then the slope of this line is $tan (theta)$. The slope of the second line is $tan (pi /2+theta)=-cot(theta)$. Since $tan (theta)$ $(-cot (theta))=-1$ we are done.
answered Mar 25 at 9:41
Kavi Rama MurthyKavi Rama Murthy
74.6k53270
$endgroup$
$begingroup$
Thanks a lot, sir.
$endgroup$
– Ramana
Mar 25 at 9:43
add a comment |
$begingroup$
If $theta$ is the angle made by the first line with $x-$ axis then the slope of this line is $tan (theta)$. The slope of the second line is $tan (pi /2+theta)=-cot(theta)$. Since $tan (theta)$ $(-cot (theta))=-1$ we are done.
answered Mar 25 at 9:41
Kavi Rama MurthyKavi Rama Murthy
74.6k53270
$endgroup$
$begingroup$
Thanks a lot, sir.
$endgroup$
– Ramana
Mar 25 at 9:43
add a comment |
$begingroup$
If $theta$ is the angle made by the first line with $x-$ axis then the slope of this line is $tan (theta)$. The slope of the second line is $tan (pi /2+theta)=-cot(theta)$. Since $tan (theta)$ $(-cot (theta))=-1$ we are done.
answered Mar 25 at 9:41
Kavi Rama MurthyKavi Rama Murthy
74.6k53270
$endgroup$
If $theta$ is the angle made by the first line with $x-$ axis then the slope of this line is $tan (theta)$. The slope of the second line is $tan (pi /2+theta)=-cot(theta)$. Since $tan (theta)$ $(-cot (theta))=-1$ we are done.
answered Mar 25 at 9:41
Kavi Rama MurthyKavi Rama Murthy
74.6k53270
answered Mar 25 at 9:41
Kavi Rama MurthyKavi Rama Murthy
74.6k53270
answered Mar 25 at 9:41
Kavi Rama MurthyKavi Rama Murthy
74.6k53270
answered Mar 25 at 9:41
Kavi Rama MurthyKavi Rama Murthy
74.6k53270
74.6k53270
$begingroup$
Thanks a lot, sir.
$endgroup$
– Ramana
Mar 25 at 9:43
add a comment |
$begingroup$
Thanks a lot, sir.
$endgroup$
– Ramana
Mar 25 at 9:43
$begingroup$
Thanks a lot, sir.
$endgroup$
– Ramana
Mar 25 at 9:43
$begingroup$
Thanks a lot, sir.
$endgroup$
– Ramana
Mar 25 at 9:43
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3161568%2fusing-trigonometric-formulas-to-prove-that-m-1m-2-1-for-perpendicular-lines%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
3
$begingroup$
If you know vectors, you can recall that the lines have direction vectors $beginpmatrix1\ m_1endpmatrix$ and $beginpmatrix1\ m_2endpmatrix$ respectively. Then note that the lines are perpendicular if and only if their directions vectors are perpendicular (i.e. their dot product is $0$). This will get you the result.
$endgroup$
– Minus One-Twelfth
Mar 25 at 9:43
$begingroup$
Minus One-Twelfth.Right your comment as an snsert , I delete mine.OK?
$endgroup$
– Peter Szilas
Mar 25 at 10:04