What does the dot product give us?
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I am working through some questions and I have an equation for the position vector of a projectile, namely r, and it's given that k is the vector directly vertical to the starting point of the projectile. Why is it that when you calculate the dot product of r and k (at time t when k is maximised) this gives you the maximum height of the projectile. i.e. what information is the dot product giving us.
geometry vectors physics
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add a comment |
$begingroup$
I am working through some questions and I have an equation for the position vector of a projectile, namely r, and it's given that k is the vector directly vertical to the starting point of the projectile. Why is it that when you calculate the dot product of r and k (at time t when k is maximised) this gives you the maximum height of the projectile. i.e. what information is the dot product giving us.
geometry vectors physics
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1
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Have you heard of orthogonal projections ?
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– Yves Daoust
Dec 9 '18 at 15:08
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See here: math.stackexchange.com/questions/1321964/….
$endgroup$
– Michael Hoppe
Dec 9 '18 at 15:52
add a comment |
$begingroup$
I am working through some questions and I have an equation for the position vector of a projectile, namely r, and it's given that k is the vector directly vertical to the starting point of the projectile. Why is it that when you calculate the dot product of r and k (at time t when k is maximised) this gives you the maximum height of the projectile. i.e. what information is the dot product giving us.
geometry vectors physics
$endgroup$
I am working through some questions and I have an equation for the position vector of a projectile, namely r, and it's given that k is the vector directly vertical to the starting point of the projectile. Why is it that when you calculate the dot product of r and k (at time t when k is maximised) this gives you the maximum height of the projectile. i.e. what information is the dot product giving us.
geometry vectors physics
geometry vectors physics
asked Dec 9 '18 at 15:06
user571032
1
$begingroup$
Have you heard of orthogonal projections ?
$endgroup$
– Yves Daoust
Dec 9 '18 at 15:08
$begingroup$
See here: math.stackexchange.com/questions/1321964/….
$endgroup$
– Michael Hoppe
Dec 9 '18 at 15:52
add a comment |
1
$begingroup$
Have you heard of orthogonal projections ?
$endgroup$
– Yves Daoust
Dec 9 '18 at 15:08
$begingroup$
See here: math.stackexchange.com/questions/1321964/….
$endgroup$
– Michael Hoppe
Dec 9 '18 at 15:52
1
1
$begingroup$
Have you heard of orthogonal projections ?
$endgroup$
– Yves Daoust
Dec 9 '18 at 15:08
$begingroup$
Have you heard of orthogonal projections ?
$endgroup$
– Yves Daoust
Dec 9 '18 at 15:08
$begingroup$
See here: math.stackexchange.com/questions/1321964/….
$endgroup$
– Michael Hoppe
Dec 9 '18 at 15:52
$begingroup$
See here: math.stackexchange.com/questions/1321964/….
$endgroup$
– Michael Hoppe
Dec 9 '18 at 15:52
add a comment |
1 Answer
1
active
oldest
votes
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Short answer: This goes back to Galileo Galilei, and his theory that motion can be decomposed into independent horizontal and vertical components. Scalar product is the formal machinery that algebraic geometry uses to decompose this way.
answered Dec 9 '18 at 15:14
ArthurArthur
115k7116198
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$begingroup$
How could I think of the schematically?
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– user571032
Dec 9 '18 at 15:16
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So here the dot product gives the vertical component of the position vector. Which is what you require the height.
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– mm-crj
Dec 9 '18 at 15:16
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@ojd What do you actually know about the dot product?
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– Arthur
Dec 9 '18 at 15:20
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@ojd If $k$ is vertical and has length $1$, then taking the dot product with $k$ does give you the size of the vertical component of your vector. Yes.
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– Arthur
Dec 9 '18 at 15:25
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Short answer: This goes back to Galileo Galilei, and his theory that motion can be decomposed into independent horizontal and vertical components. Scalar product is the formal machinery that algebraic geometry uses to decompose this way.
answered Dec 9 '18 at 15:14
ArthurArthur
115k7116198
$endgroup$
$begingroup$
How could I think of the schematically?
$endgroup$
– user571032
Dec 9 '18 at 15:16
$begingroup$
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
$endgroup$
– mm-crj
Dec 9 '18 at 15:16
$begingroup$
@ojd What do you actually know about the dot product?
$endgroup$
– Arthur
Dec 9 '18 at 15:20
$begingroup$
@ojd If $k$ is vertical and has length $1$, then taking the dot product with $k$ does give you the size of the vertical component of your vector. Yes.
$endgroup$
– Arthur
Dec 9 '18 at 15:25
add a comment |
$begingroup$
Short answer: This goes back to Galileo Galilei, and his theory that motion can be decomposed into independent horizontal and vertical components. Scalar product is the formal machinery that algebraic geometry uses to decompose this way.
answered Dec 9 '18 at 15:14
ArthurArthur
115k7116198
$endgroup$
$begingroup$
How could I think of the schematically?
$endgroup$
– user571032
Dec 9 '18 at 15:16
$begingroup$
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
$endgroup$
– mm-crj
Dec 9 '18 at 15:16
$begingroup$
@ojd What do you actually know about the dot product?
$endgroup$
– Arthur
Dec 9 '18 at 15:20
$begingroup$
@ojd If $k$ is vertical and has length $1$, then taking the dot product with $k$ does give you the size of the vertical component of your vector. Yes.
$endgroup$
– Arthur
Dec 9 '18 at 15:25
add a comment |
$begingroup$
Short answer: This goes back to Galileo Galilei, and his theory that motion can be decomposed into independent horizontal and vertical components. Scalar product is the formal machinery that algebraic geometry uses to decompose this way.
answered Dec 9 '18 at 15:14
ArthurArthur
115k7116198
$endgroup$
Short answer: This goes back to Galileo Galilei, and his theory that motion can be decomposed into independent horizontal and vertical components. Scalar product is the formal machinery that algebraic geometry uses to decompose this way.
answered Dec 9 '18 at 15:14
ArthurArthur
115k7116198
answered Dec 9 '18 at 15:14
ArthurArthur
115k7116198
answered Dec 9 '18 at 15:14
ArthurArthur
115k7116198
answered Dec 9 '18 at 15:14
ArthurArthur
115k7116198
115k7116198
$begingroup$
How could I think of the schematically?
$endgroup$
– user571032
Dec 9 '18 at 15:16
$begingroup$
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
$endgroup$
– mm-crj
Dec 9 '18 at 15:16
$begingroup$
@ojd What do you actually know about the dot product?
$endgroup$
– Arthur
Dec 9 '18 at 15:20
$begingroup$
@ojd If $k$ is vertical and has length $1$, then taking the dot product with $k$ does give you the size of the vertical component of your vector. Yes.
$endgroup$
– Arthur
Dec 9 '18 at 15:25
add a comment |
$begingroup$
How could I think of the schematically?
$endgroup$
– user571032
Dec 9 '18 at 15:16
$begingroup$
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
$endgroup$
– mm-crj
Dec 9 '18 at 15:16
$begingroup$
@ojd What do you actually know about the dot product?
$endgroup$
– Arthur
Dec 9 '18 at 15:20
$begingroup$
@ojd If $k$ is vertical and has length $1$, then taking the dot product with $k$ does give you the size of the vertical component of your vector. Yes.
$endgroup$
– Arthur
Dec 9 '18 at 15:25
$begingroup$
How could I think of the schematically?
$endgroup$
– user571032
Dec 9 '18 at 15:16
$begingroup$
How could I think of the schematically?
$endgroup$
– user571032
Dec 9 '18 at 15:16
$begingroup$
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
$endgroup$
– mm-crj
Dec 9 '18 at 15:16
$begingroup$
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
$endgroup$
– mm-crj
Dec 9 '18 at 15:16
$begingroup$
@ojd What do you actually know about the dot product?
$endgroup$
– Arthur
Dec 9 '18 at 15:20
$begingroup$
@ojd What do you actually know about the dot product?
$endgroup$
– Arthur
Dec 9 '18 at 15:20
$begingroup$
@ojd If $k$ is vertical and has length $1$, then taking the dot product with $k$ does give you the size of the vertical component of your vector. Yes.
$endgroup$
– Arthur
Dec 9 '18 at 15:25
$begingroup$
@ojd If $k$ is vertical and has length $1$, then taking the dot product with $k$ does give you the size of the vertical component of your vector. Yes.
$endgroup$
– Arthur
Dec 9 '18 at 15:25
add a comment |
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1
$begingroup$
Have you heard of orthogonal projections ?
$endgroup$
– Yves Daoust
Dec 9 '18 at 15:08
$begingroup$
See here: math.stackexchange.com/questions/1321964/….
$endgroup$
– Michael Hoppe
Dec 9 '18 at 15:52