Vector projection vs Scalar projection | Mathematics | Page 1

violet fable Jun 4, 2024, 9:04 AM

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Hi,
I've been looking at the vector projection, and I've been struggling to find how these 2 are related. Could anyone please explain?

humble flameBOT Jun 4, 2024, 9:04 AM

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fresh crag Jun 4, 2024, 9:12 AM

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the scalar projection is the length of the vector projection

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i disagree with how they write the vector projection

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i usually write it as

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$\frac{\vec{a}\cdot\vec{b}}{\norm{\vec{b}}^2}\vec{b}$

keen hullBOT Jun 4, 2024, 9:13 AM

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elrichardo1337

fresh crag Jun 4, 2024, 9:14 AM

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now $\frac{\vec{b}}{\norm{\vec{b}}}$ is the unit vector pointing in the direciton of $\vec{b}$

keen hullBOT Jun 4, 2024, 9:14 AM

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elrichardo1337

fresh crag Jun 4, 2024, 9:14 AM

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so if you pull that factor out of the vector projection we have

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$\frac{\vec{a}\cdot\vec{b}}{\norm{\vec{b}}}\left(\frac{\vec{b}}{\norm{\vec{b}}}\right)$

keen hullBOT Jun 4, 2024, 9:15 AM

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elrichardo1337

fresh crag Jun 4, 2024, 9:15 AM

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the stuff outside the parentheses is a scalar, the vector in the parentheses has unit length

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so the stuff outside, the length of the vector projection, is your scalar projection

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hope that makes sense

violet fable Jun 4, 2024, 9:18 AM

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hmm

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let me think about it for a minute!

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so the vector projection is made of the scalar projection (the magnitude, or dot product) and something else which is the direction?

fresh crag Jun 4, 2024, 9:24 AM

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yeah basically

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the direction part is the unit vector i mentioned

violet fable Jun 4, 2024, 9:24 AM

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oh the unit vector thats the name

fresh crag Jun 4, 2024, 9:24 AM

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yea

violet fable Jun 4, 2024, 9:25 AM

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tysm

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i gotta revise some sequences and series and then i have my exam tomorrow :(

fresh crag Jun 4, 2024, 9:26 AM

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np

#Vector projection vs Scalar projection