How do I find an equation from vectors of a certain form?
$begingroup$
Consider the plane $P$ in $mathbb{R}^3$ containing all the vectors of the form:
$r begin{bmatrix} 1 \ 2 \ 3 end{bmatrix} + s begin{bmatrix} 3 \ 2 \ 1 end{bmatrix}
$
Find an equation which describes the plane and show the point $(5,6,5)$ is not on the plane $P$.
linear-algebra
edited Jan 15 at 23:04
jwc845
382215
asked Jan 15 at 22:01
Brooklyn Van HellBrooklyn Van Hell
31
$endgroup$
add a comment |
$begingroup$
Consider the plane $P$ in $mathbb{R}^3$ containing all the vectors of the form:
$r begin{bmatrix} 1 \ 2 \ 3 end{bmatrix} + s begin{bmatrix} 3 \ 2 \ 1 end{bmatrix}
$
Find an equation which describes the plane and show the point $(5,6,5)$ is not on the plane $P$.
linear-algebra
edited Jan 15 at 23:04
jwc845
382215
asked Jan 15 at 22:01
Brooklyn Van HellBrooklyn Van Hell
31
$endgroup$
2
$begingroup$
What have you tried so far?
$endgroup$
– jwc845
Jan 15 at 22:11
add a comment |
$begingroup$
Consider the plane $P$ in $mathbb{R}^3$ containing all the vectors of the form:
$r begin{bmatrix} 1 \ 2 \ 3 end{bmatrix} + s begin{bmatrix} 3 \ 2 \ 1 end{bmatrix}
$
Find an equation which describes the plane and show the point $(5,6,5)$ is not on the plane $P$.
linear-algebra
edited Jan 15 at 23:04
jwc845
382215
asked Jan 15 at 22:01
Brooklyn Van HellBrooklyn Van Hell
31
$endgroup$
Consider the plane $P$ in $mathbb{R}^3$ containing all the vectors of the form:
$r begin{bmatrix} 1 \ 2 \ 3 end{bmatrix} + s begin{bmatrix} 3 \ 2 \ 1 end{bmatrix}
$
Find an equation which describes the plane and show the point $(5,6,5)$ is not on the plane $P$.
linear-algebra
linear-algebra
edited Jan 15 at 23:04
jwc845
382215
asked Jan 15 at 22:01
Brooklyn Van HellBrooklyn Van Hell
31
edited Jan 15 at 23:04
jwc845
382215
asked Jan 15 at 22:01
Brooklyn Van HellBrooklyn Van Hell
31
edited Jan 15 at 23:04
jwc845
382215
edited Jan 15 at 23:04
jwc845
382215
edited Jan 15 at 23:04
jwc845
382215
382215
asked Jan 15 at 22:01
Brooklyn Van HellBrooklyn Van Hell
31
asked Jan 15 at 22:01
Brooklyn Van HellBrooklyn Van Hell
31
asked Jan 15 at 22:01
Brooklyn Van HellBrooklyn Van Hell
31
31
2
$begingroup$
What have you tried so far?
$endgroup$
– jwc845
Jan 15 at 22:11
add a comment |
2
$begingroup$
What have you tried so far?
$endgroup$
– jwc845
Jan 15 at 22:11
2
2
$begingroup$
What have you tried so far?
$endgroup$
– jwc845
Jan 15 at 22:11
$begingroup$
What have you tried so far?
$endgroup$
– jwc845
Jan 15 at 22:11
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
You can take the cross-product to get the normal: $(1,2,3)×(3,2,1)=begin{vmatrix}i&j&k\1&2&3\3&2&1end{vmatrix}=-4i+8j-4k$.
Then use the point $(1,2,3)$: $4x-8y+4z=dimplies 4(1)-8(2)+3(4)=0=d$. $d$ should be zero, because the plane goes through the origin.
So, $4x-8y+4z=0$ would be an equation of the plane.
Lastly, $4(5)-8(6)+4(5)=-8neq0$. So $(5,6,5)$ is not on the plane.
answered Jan 15 at 22:13
Chris CusterChris Custer
14.6k3827
$endgroup$
$begingroup$
This helped a lot! Thanks so much!
$endgroup$
– Brooklyn Van Hell
Jan 15 at 22:25
$begingroup$
You're welcome.
$endgroup$
– Chris Custer
Jan 15 at 23:08
add a comment |
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1 Answer
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oldest
votes
1 Answer
1
active
oldest
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$begingroup$
You can take the cross-product to get the normal: $(1,2,3)×(3,2,1)=begin{vmatrix}i&j&k\1&2&3\3&2&1end{vmatrix}=-4i+8j-4k$.
Then use the point $(1,2,3)$: $4x-8y+4z=dimplies 4(1)-8(2)+3(4)=0=d$. $d$ should be zero, because the plane goes through the origin.
So, $4x-8y+4z=0$ would be an equation of the plane.
Lastly, $4(5)-8(6)+4(5)=-8neq0$. So $(5,6,5)$ is not on the plane.
answered Jan 15 at 22:13
Chris CusterChris Custer
14.6k3827
$endgroup$
$begingroup$
This helped a lot! Thanks so much!
$endgroup$
– Brooklyn Van Hell
Jan 15 at 22:25
$begingroup$
You're welcome.
$endgroup$
– Chris Custer
Jan 15 at 23:08
add a comment |
$begingroup$
You can take the cross-product to get the normal: $(1,2,3)×(3,2,1)=begin{vmatrix}i&j&k\1&2&3\3&2&1end{vmatrix}=-4i+8j-4k$.
Then use the point $(1,2,3)$: $4x-8y+4z=dimplies 4(1)-8(2)+3(4)=0=d$. $d$ should be zero, because the plane goes through the origin.
So, $4x-8y+4z=0$ would be an equation of the plane.
Lastly, $4(5)-8(6)+4(5)=-8neq0$. So $(5,6,5)$ is not on the plane.
answered Jan 15 at 22:13
Chris CusterChris Custer
14.6k3827
$endgroup$
$begingroup$
This helped a lot! Thanks so much!
$endgroup$
– Brooklyn Van Hell
Jan 15 at 22:25
$begingroup$
You're welcome.
$endgroup$
– Chris Custer
Jan 15 at 23:08
add a comment |
$begingroup$
You can take the cross-product to get the normal: $(1,2,3)×(3,2,1)=begin{vmatrix}i&j&k\1&2&3\3&2&1end{vmatrix}=-4i+8j-4k$.
Then use the point $(1,2,3)$: $4x-8y+4z=dimplies 4(1)-8(2)+3(4)=0=d$. $d$ should be zero, because the plane goes through the origin.
So, $4x-8y+4z=0$ would be an equation of the plane.
Lastly, $4(5)-8(6)+4(5)=-8neq0$. So $(5,6,5)$ is not on the plane.
answered Jan 15 at 22:13
Chris CusterChris Custer
14.6k3827
$endgroup$
You can take the cross-product to get the normal: $(1,2,3)×(3,2,1)=begin{vmatrix}i&j&k\1&2&3\3&2&1end{vmatrix}=-4i+8j-4k$.
Then use the point $(1,2,3)$: $4x-8y+4z=dimplies 4(1)-8(2)+3(4)=0=d$. $d$ should be zero, because the plane goes through the origin.
So, $4x-8y+4z=0$ would be an equation of the plane.
Lastly, $4(5)-8(6)+4(5)=-8neq0$. So $(5,6,5)$ is not on the plane.
answered Jan 15 at 22:13
Chris CusterChris Custer
14.6k3827
edited Jan 15 at 22:37
answered Jan 15 at 22:13
Chris CusterChris Custer
14.6k3827
answered Jan 15 at 22:13
Chris CusterChris Custer
14.6k3827
answered Jan 15 at 22:13
Chris CusterChris Custer
14.6k3827
14.6k3827
$begingroup$
This helped a lot! Thanks so much!
$endgroup$
– Brooklyn Van Hell
Jan 15 at 22:25
$begingroup$
You're welcome.
$endgroup$
– Chris Custer
Jan 15 at 23:08
add a comment |
$begingroup$
This helped a lot! Thanks so much!
$endgroup$
– Brooklyn Van Hell
Jan 15 at 22:25
$begingroup$
You're welcome.
$endgroup$
– Chris Custer
Jan 15 at 23:08
$begingroup$
This helped a lot! Thanks so much!
$endgroup$
– Brooklyn Van Hell
Jan 15 at 22:25
$begingroup$
This helped a lot! Thanks so much!
$endgroup$
– Brooklyn Van Hell
Jan 15 at 22:25
$begingroup$
You're welcome.
$endgroup$
– Chris Custer
Jan 15 at 23:08
$begingroup$
You're welcome.
$endgroup$
– Chris Custer
Jan 15 at 23:08
add a comment |
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$begingroup$
What have you tried so far?
$endgroup$
– jwc845
Jan 15 at 22:11