What does the dot product give us?
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
asked Dec 9 at 15:06
ojd
696
add a comment |
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
asked Dec 9 at 15:06
ojd
696
1
Have you heard of orthogonal projections ?
– Yves Daoust
Dec 9 at 15:08
See here: math.stackexchange.com/questions/1321964/….
– Michael Hoppe
Dec 9 at 15:52
add a comment |
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
asked Dec 9 at 15:06
ojd
696
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 at 15:06
ojd
696
asked Dec 9 at 15:06
ojd
696
asked Dec 9 at 15:06
ojd
696
asked Dec 9 at 15:06
ojd
696
asked Dec 9 at 15:06
ojd
696
696
1
Have you heard of orthogonal projections ?
– Yves Daoust
Dec 9 at 15:08
See here: math.stackexchange.com/questions/1321964/….
– Michael Hoppe
Dec 9 at 15:52
add a comment |
1
Have you heard of orthogonal projections ?
– Yves Daoust
Dec 9 at 15:08
See here: math.stackexchange.com/questions/1321964/….
– Michael Hoppe
Dec 9 at 15:52
1
1
Have you heard of orthogonal projections ?
– Yves Daoust
Dec 9 at 15:08
Have you heard of orthogonal projections ?
– Yves Daoust
Dec 9 at 15:08
See here: math.stackexchange.com/questions/1321964/….
– Michael Hoppe
Dec 9 at 15:52
See here: math.stackexchange.com/questions/1321964/….
– Michael Hoppe
Dec 9 at 15:52
add a comment |
1 Answer
1
active
oldest
votes
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 at 15:14
Arthur
110k7105186
How could I think of the schematically?
– ojd
Dec 9 at 15:16
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
– mm-crj
Dec 9 at 15:16
@ojd What do you actually know about the dot product?
– Arthur
Dec 9 at 15:20
@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.
– Arthur
Dec 9 at 15:25
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 at 15:14
Arthur
110k7105186
How could I think of the schematically?
– ojd
Dec 9 at 15:16
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
– mm-crj
Dec 9 at 15:16
@ojd What do you actually know about the dot product?
– Arthur
Dec 9 at 15:20
@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.
– Arthur
Dec 9 at 15:25
add a comment |
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 at 15:14
Arthur
110k7105186
How could I think of the schematically?
– ojd
Dec 9 at 15:16
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
– mm-crj
Dec 9 at 15:16
@ojd What do you actually know about the dot product?
– Arthur
Dec 9 at 15:20
@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.
– Arthur
Dec 9 at 15:25
add a comment |
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 at 15:14
Arthur
110k7105186
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 at 15:14
Arthur
110k7105186
answered Dec 9 at 15:14
Arthur
110k7105186
answered Dec 9 at 15:14
Arthur
110k7105186
answered Dec 9 at 15:14
Arthur
110k7105186
110k7105186
How could I think of the schematically?
– ojd
Dec 9 at 15:16
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
– mm-crj
Dec 9 at 15:16
@ojd What do you actually know about the dot product?
– Arthur
Dec 9 at 15:20
@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.
– Arthur
Dec 9 at 15:25
add a comment |
How could I think of the schematically?
– ojd
Dec 9 at 15:16
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
– mm-crj
Dec 9 at 15:16
@ojd What do you actually know about the dot product?
– Arthur
Dec 9 at 15:20
@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.
– Arthur
Dec 9 at 15:25
How could I think of the schematically?
– ojd
Dec 9 at 15:16
How could I think of the schematically?
– ojd
Dec 9 at 15:16
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
– mm-crj
Dec 9 at 15:16
So here the dot product gives the vertical component of the position vector. Which is what you require the height.
– mm-crj
Dec 9 at 15:16
@ojd What do you actually know about the dot product?
– Arthur
Dec 9 at 15:20
@ojd What do you actually know about the dot product?
– Arthur
Dec 9 at 15:20
@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.
– Arthur
Dec 9 at 15:25
@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.
– Arthur
Dec 9 at 15:25
add a comment |
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1
Have you heard of orthogonal projections ?
– Yves Daoust
Dec 9 at 15:08
See here: math.stackexchange.com/questions/1321964/….
– Michael Hoppe
Dec 9 at 15:52