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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  • 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
















0












$begingroup$


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







share|cite|improve this question









$endgroup$








  • 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














0












0








0


0



$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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














  • 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










1 Answer
1






active

oldest

votes


















1












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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













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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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


















1












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
















1












1








1





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 9 '18 at 15:14









ArthurArthur

114k7115197




114k7115197












  • $begingroup$
    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










  • $begingroup$
    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




















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