basic_tools:vector_calculus:dot_product
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basic_tools:vector_calculus:dot_product [2017/12/16 14:51] jakobadmin [Student] |
basic_tools:vector_calculus:dot_product [2018/03/28 12:26] (current) jakobadmin |
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| - | ====== Dot Product ====== | + | ====== Dot Product / Scalar Product ====== |
| - | <tabbox Why is it interesting?> | ||
| - | <tabbox Layman> | + | <tabbox Intuitive> |
| <note tip> | <note tip> | ||
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| </note> | </note> | ||
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| - | <tabbox Student> | + | <tabbox Concrete> |
| + | {{ :basic_tools:vector_calculus:dotproduct.png?nolink&200|}} | ||
| <blockquote>The [dot] product may be understood geometrically as the **projection** of one vector onto another, multiplied by the length of the vector that it is projected onto. If one takes the dot product of two vectors $\vec{a}$ and $\vec{b}$, we can apply this procedure to find the correct formula for the dot product: | <blockquote>The [dot] product may be understood geometrically as the **projection** of one vector onto another, multiplied by the length of the vector that it is projected onto. If one takes the dot product of two vectors $\vec{a}$ and $\vec{b}$, we can apply this procedure to find the correct formula for the dot product: | ||
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| This is the correct expression for the dot product. Of course, the dot product is symmetric so we might as well picture it as projecting $\vec{b}$ along $\vec{a}$. | This is the correct expression for the dot product. Of course, the dot product is symmetric so we might as well picture it as projecting $\vec{b}$ along $\vec{a}$. | ||
| <cite>https://physics.stackexchange.com/a/111869/37286</cite></blockquote> | <cite>https://physics.stackexchange.com/a/111869/37286</cite></blockquote> | ||
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| * [[https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/|Vector Calculus: Understanding the Dot Product]] by Kalid Azad | * [[https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/|Vector Calculus: Understanding the Dot Product]] by Kalid Azad | ||
| * See also https://math.stackexchange.com/questions/348717/dot-product-intuition | * See also https://math.stackexchange.com/questions/348717/dot-product-intuition | ||
| - | <tabbox Researcher> | + | <tabbox Abstract> |
| <note tip> | <note tip> | ||
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| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| - | --> Example1# | + | The dot product is a tool that we can use to combine two vectors and get a number out. (That's why it is also called scalar product; scalar=number). This number tells us how much the first vector points in the direction of the second vector. |
| - | + | This is an extremely useful concept and used in almost any physical theory, like for example, [[models:classical_electrodynamics|electrodynamics]]. Moreover, many other important tools, like the [[basic_tools:vector_calculus:divergence|divergence]] or the [[basic_tools:vector_calculus:gradient|gradient]] are defined with the help of the dot product. | |
| - | <-- | + | |
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| - | --> Example2:# | + | |
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| - | <-- | + | |
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| - | <tabbox FAQ> | + | |
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| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
basic_tools/vector_calculus/dot_product.1513432294.txt.gz · Last modified: 2017/12/16 13:51 (external edit)
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