Mixing
Show that OC is parallel to $left(fraca+fracbbright)$
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The points O, A and B are on a plane such that relative to the point O, the points A and B have
non-parallel position vectors a and b respectively.
The point C with position vector $c$ is on the plane OAB such that OC bisects the angle AOB. The question asks me to show $$left(fraca-fracbright)cdot c = 0$$ and i can show it easily by using the formula for angles between two vectors.
But next part the question says show that OC is parallel to $left(fraca+fracbright)$
I know this is true as I know the unit vector $a$ plus unit vector $b$ must be in the direction of $c$, but I am not sure how to show it rigorously using theorems.
geometry vectors
asked Jul 25 at 14:20
ilovewt
821314
add a comment |Â
up vote
2
down vote
favorite
The points O, A and B are on a plane such that relative to the point O, the points A and B have
non-parallel position vectors a and b respectively.
The point C with position vector $c$ is on the plane OAB such that OC bisects the angle AOB. The question asks me to show $$left(fraca-fracbright)cdot c = 0$$ and i can show it easily by using the formula for angles between two vectors.
But next part the question says show that OC is parallel to $left(fraca+fracbright)$
I know this is true as I know the unit vector $a$ plus unit vector $b$ must be in the direction of $c$, but I am not sure how to show it rigorously using theorems.
geometry vectors
asked Jul 25 at 14:20
ilovewt
821314
1
Think of the parallelogram rule for 2 vectors.
– Mythomorphic
Jul 25 at 14:26
@Mythomorphic Yes I did but how can I confirm that $a+b$ will give me a resultant in the direction of $c$?
– ilovewt
Jul 25 at 14:29
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The points O, A and B are on a plane such that relative to the point O, the points A and B have
non-parallel position vectors a and b respectively.
The point C with position vector $c$ is on the plane OAB such that OC bisects the angle AOB. The question asks me to show $$left(fraca-fracbright)cdot c = 0$$ and i can show it easily by using the formula for angles between two vectors.
But next part the question says show that OC is parallel to $left(fraca+fracbright)$
I know this is true as I know the unit vector $a$ plus unit vector $b$ must be in the direction of $c$, but I am not sure how to show it rigorously using theorems.
geometry vectors
asked Jul 25 at 14:20
ilovewt
821314
The points O, A and B are on a plane such that relative to the point O, the points A and B have
non-parallel position vectors a and b respectively.
The point C with position vector $c$ is on the plane OAB such that OC bisects the angle AOB. The question asks me to show $$left(fraca-fracbright)cdot c = 0$$ and i can show it easily by using the formula for angles between two vectors.
But next part the question says show that OC is parallel to $left(fraca+fracbright)$
I know this is true as I know the unit vector $a$ plus unit vector $b$ must be in the direction of $c$, but I am not sure how to show it rigorously using theorems.
geometry vectors
asked Jul 25 at 14:20
ilovewt
821314
asked Jul 25 at 14:20
ilovewt
821314
asked Jul 25 at 14:20
ilovewt
821314
asked Jul 25 at 14:20
ilovewt
821314
821314
1
Think of the parallelogram rule for 2 vectors.
– Mythomorphic
Jul 25 at 14:26
@Mythomorphic Yes I did but how can I confirm that $a+b$ will give me a resultant in the direction of $c$?
– ilovewt
Jul 25 at 14:29
add a comment |Â
1
Think of the parallelogram rule for 2 vectors.
– Mythomorphic
Jul 25 at 14:26
@Mythomorphic Yes I did but how can I confirm that $a+b$ will give me a resultant in the direction of $c$?
– ilovewt
Jul 25 at 14:29
1
1
Think of the parallelogram rule for 2 vectors.
– Mythomorphic
Jul 25 at 14:26
Think of the parallelogram rule for 2 vectors.
– Mythomorphic
Jul 25 at 14:26
@Mythomorphic Yes I did but how can I confirm that $a+b$ will give me a resultant in the direction of $c$?
– ilovewt
Jul 25 at 14:29
@Mythomorphic Yes I did but how can I confirm that $a+b$ will give me a resultant in the direction of $c$?
– ilovewt
Jul 25 at 14:29
add a comment |Â
1 Answer
1
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votes
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1
down vote
accepted
First we show that if $a$ and $b$ are unit vectors, $a+b$ and $a-b$ are orthogonal to each other. The proof is simple: $(a+b)cdot(a-b)=acdot a-acdot b+bcdot a-bcdot b=Vert aVert^2-Vert bVert^2=1-1=0$.
The first part of the question gives that $c$ is perpendicular to $frac a-frac b$, which is itself perpendicular to $frac a+frac b$ by the lemma above. Thus, since we are working in a plane, $c$ is parallel to $frac a+frac b$ as required.
answered Jul 25 at 14:40
Parcly Taxel
33.5k136588
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
First we show that if $a$ and $b$ are unit vectors, $a+b$ and $a-b$ are orthogonal to each other. The proof is simple: $(a+b)cdot(a-b)=acdot a-acdot b+bcdot a-bcdot b=Vert aVert^2-Vert bVert^2=1-1=0$.
The first part of the question gives that $c$ is perpendicular to $frac a-frac b$, which is itself perpendicular to $frac a+frac b$ by the lemma above. Thus, since we are working in a plane, $c$ is parallel to $frac a+frac b$ as required.
answered Jul 25 at 14:40
Parcly Taxel
33.5k136588
add a comment |Â
up vote
1
down vote
accepted
First we show that if $a$ and $b$ are unit vectors, $a+b$ and $a-b$ are orthogonal to each other. The proof is simple: $(a+b)cdot(a-b)=acdot a-acdot b+bcdot a-bcdot b=Vert aVert^2-Vert bVert^2=1-1=0$.
The first part of the question gives that $c$ is perpendicular to $frac a-frac b$, which is itself perpendicular to $frac a+frac b$ by the lemma above. Thus, since we are working in a plane, $c$ is parallel to $frac a+frac b$ as required.
answered Jul 25 at 14:40
Parcly Taxel
33.5k136588
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
First we show that if $a$ and $b$ are unit vectors, $a+b$ and $a-b$ are orthogonal to each other. The proof is simple: $(a+b)cdot(a-b)=acdot a-acdot b+bcdot a-bcdot b=Vert aVert^2-Vert bVert^2=1-1=0$.
The first part of the question gives that $c$ is perpendicular to $frac a-frac b$, which is itself perpendicular to $frac a+frac b$ by the lemma above. Thus, since we are working in a plane, $c$ is parallel to $frac a+frac b$ as required.
answered Jul 25 at 14:40
Parcly Taxel
33.5k136588
First we show that if $a$ and $b$ are unit vectors, $a+b$ and $a-b$ are orthogonal to each other. The proof is simple: $(a+b)cdot(a-b)=acdot a-acdot b+bcdot a-bcdot b=Vert aVert^2-Vert bVert^2=1-1=0$.
The first part of the question gives that $c$ is perpendicular to $frac a-frac b$, which is itself perpendicular to $frac a+frac b$ by the lemma above. Thus, since we are working in a plane, $c$ is parallel to $frac a+frac b$ as required.
answered Jul 25 at 14:40
Parcly Taxel
33.5k136588
answered Jul 25 at 14:40
Parcly Taxel
33.5k136588
answered Jul 25 at 14:40
Parcly Taxel
33.5k136588
answered Jul 25 at 14:40
Parcly Taxel
33.5k136588
33.5k136588
add a comment |Â
add a comment |Â
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1
Think of the parallelogram rule for 2 vectors.
– Mythomorphic
Jul 25 at 14:26
@Mythomorphic Yes I did but how can I confirm that $a+b$ will give me a resultant in the direction of $c$?
– ilovewt
Jul 25 at 14:29