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diff --git a/build/html/Mathematics/Algebra/Logarithm.html b/build/html/Mathematics/Algebra/Logarithm.html new file mode 100644 index 0000000..9d1ffe2 --- /dev/null +++ b/build/html/Mathematics/Algebra/Logarithm.html @@ -0,0 +1,435 @@ + + + + + + + + + + + Logarithm — Neo's Journal + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Logarithm

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Contents

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Logarithm#

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Logarithm Rules#

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Product Rule

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Hello

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Algebra

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Linear Approximation

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Logarithm

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Mathematics

Mathematics#

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      Contents

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      Normalising a vector#

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      Normalising a vector is the process of making the length +of a vector equal to 1, while maintaining the direction. +This is called a unit vector.

      +
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      Formula#

      +
      +\[\overrightarrow{u} = { \overrightarrow{v} \over {\vert\vert\overrightarrow{v} \vert\vert} }\]
      +
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      Intuition#

      +

      Building an understanding of this process. +Starting out with a 2D vector \(\lt2, 4\gt\).

      +../_images/normalisevectors1.png +

      We can compute the hypotenuse with Pythagoras.

      +

      \(\sqrt{2^2 + 4^2} = \sqrt{20}\)

      +

      So the question then becomes, “What can I do to \(\sqrt{20}\) +to make it equal to 1”. That’s an easy question to answer, just +divide it by itself. BUT the catch is preserving the angle.

      +

      Working out the angle we get \(\arcsin({4 \over \sqrt{20}}) \approx 63.43\)

      +

      Our unit vector will always be a length of 1. That means our vector point +will lie at 63.43 degrees on a unit circle.

      +../_images/normalisevectors2.png +

      To get our values we just plug in 63.43 into cosine and sine.

      +
      +\[ \begin{align}\begin{aligned}\sin{(63.43)} = 0.89\\\cos{(63.43)} = 0.45\end{aligned}\end{align} \]
      +

      Which is just our values of \({4 \over \sqrt{20}}\) and \({2 \over \sqrt{20}}\).

      +

      Putting it all together we can say

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      Normalising a vector#

      +

      Normalising a vector is the process of making the length +of a vector equal to 1, while maintaining the direction. +This is called a unit vector.

      +
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      diff --git a/build/html/_images/normalisevectors1.png b/build/html/_images/normalisevectors1.png new file mode 100644 index 0000000..d0342f4 Binary files /dev/null and b/build/html/_images/normalisevectors1.png differ diff --git a/build/html/_images/normalisevectors2.png b/build/html/_images/normalisevectors2.png new file mode 100644 index 0000000..b23c585 Binary files /dev/null and b/build/html/_images/normalisevectors2.png differ diff --git a/build/html/_sources/Mathematics/Algebra/Logarithm.rst b/build/html/_sources/Mathematics/Algebra/Logarithm.rst new file mode 100644 index 0000000..cb8d2c0 --- /dev/null +++ b/build/html/_sources/Mathematics/Algebra/Logarithm.rst @@ -0,0 +1,10 @@ +Logarithm +========= + +Logarithm Rules +*************** + +Product Rule + +.. math:: + log_b(X \cdot Y) = log_b(X) + log_b(Y) diff --git a/build/html/_sources/Mathematics/Algebra/index.rst b/build/html/_sources/Mathematics/Algebra/index.rst index 637cb67..c4472c8 100644 --- a/build/html/_sources/Mathematics/Algebra/index.rst +++ b/build/html/_sources/Mathematics/Algebra/index.rst @@ -1,2 +1,7 @@ -Hello -====== +Algebra +******* + +.. toctree:: + :glob: + + Logarithm.rst diff --git a/build/html/_sources/Physics/index.rst b/build/html/_sources/Physics/index.rst new file mode 100644 index 0000000..e255ad0 --- /dev/null +++ b/build/html/_sources/Physics/index.rst @@ -0,0 +1,7 @@ +Physics +======= + +.. toctree:: + + Physics/index.rst + normalisevector.rst diff --git a/build/html/_sources/Physics/normalisevector.rst b/build/html/_sources/Physics/normalisevector.rst new file mode 100644 index 0000000..811171d --- /dev/null +++ b/build/html/_sources/Physics/normalisevector.rst @@ -0,0 +1,52 @@ +.. _normalisinvector: + +Normalising a vector +==================== + +Normalising a vector is the process of making the length +of a vector equal to 1, while maintaining the direction. +This is called a `unit vector`. + +Formula +******* + +.. math:: + \overrightarrow{u} = { \overrightarrow{v} \over {\vert\vert\overrightarrow{v} \vert\vert} } + + +Intuition +********* + +Building an understanding of this process. +Starting out with a 2D vector :math:`\lt2, 4\gt`. + +.. image:: /images/normalisevectors1.png + +We can compute the hypotenuse with Pythagoras. + +:math:`\sqrt{2^2 + 4^2} = \sqrt{20}` + +So the question then becomes, "What can I do to :math:`\sqrt{20}` +to make it equal to 1". That's an easy question to answer, just +divide it by itself. **BUT** the catch is preserving the angle. + +Working out the angle we get :math:`\arcsin({4 \over \sqrt{20}}) \approx 63.43` + +Our `unit vector` will always be a length of 1. That means our vector point +will lie at 63.43 degrees on a unit circle. + +.. image:: /images/normalisevectors2.png + +To get our values we just plug in 63.43 into cosine and sine. + +.. math:: + \sin{(63.43)} = 0.89 + + \cos{(63.43)} = 0.45 + +Which is just our values of :math:`{4 \over \sqrt{20}}` and :math:`{2 \over \sqrt{20}}`. + +Putting it all together we can say + +.. math:: + {{\lt2,4\gt} \over \sqrt{20}} = \lt0.45, 0.89\gt diff --git a/build/html/_sources/Physics/normalizevector.rst b/build/html/_sources/Physics/normalizevector.rst new file mode 100644 index 0000000..905da06 --- /dev/null +++ b/build/html/_sources/Physics/normalizevector.rst @@ -0,0 +1,6 @@ +Normalising a vector +==================== + +Normalising a vector is the process of making the length +of a vector equal to 1, while maintaining the direction. +This is called a `unit vector`. diff --git a/build/html/_sources/index.rst b/build/html/_sources/index.rst index 573fb2a..1cf2d9f 100644 --- a/build/html/_sources/index.rst +++ b/build/html/_sources/index.rst @@ -5,3 +5,4 @@ Neo's Journal :glob: Mathematics/index.rst + Physics/index.rst diff --git a/build/html/_static/normalisevectors1.png b/build/html/_static/normalisevectors1.png new file mode 100644 index 0000000..d0342f4 Binary files /dev/null and b/build/html/_static/normalisevectors1.png differ diff --git a/build/html/_static/normalisevectors1.svg b/build/html/_static/normalisevectors1.svg new file mode 100644 index 0000000..c0c354c --- /dev/null +++ b/build/html/_static/normalisevectors1.svg @@ -0,0 +1,93 @@ + + + + + + + + + + + + + + + + + + + + diff --git a/build/html/_static/normalisevectors2.svg b/build/html/_static/normalisevectors2.svg new file mode 100644 index 0000000..7d462a5 --- /dev/null +++ b/build/html/_static/normalisevectors2.svg @@ -0,0 +1,53 @@ + + + + + + + + + + + diff --git a/build/html/genindex.html b/build/html/genindex.html index ffa6664..e4ae2b3 100644 --- a/build/html/genindex.html +++ b/build/html/genindex.html @@ -149,10 +149,17 @@
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Rule + +.. math:: + log_b(X \cdot Y) = log_b(X) + log_b(Y) diff --git a/source/Mathematics/Algebra/index.rst b/source/Mathematics/Algebra/index.rst index 637cb67..c4472c8 100644 --- a/source/Mathematics/Algebra/index.rst +++ b/source/Mathematics/Algebra/index.rst @@ -1,2 +1,7 @@ -Hello -====== +Algebra +******* + +.. toctree:: + :glob: + + Logarithm.rst diff --git a/source/Physics/index.rst b/source/Physics/index.rst new file mode 100644 index 0000000..e255ad0 --- /dev/null +++ b/source/Physics/index.rst @@ -0,0 +1,7 @@ +Physics +======= + +.. toctree:: + + Physics/index.rst + normalisevector.rst diff --git a/source/Physics/normalisevector.rst b/source/Physics/normalisevector.rst new file mode 100644 index 0000000..811171d --- /dev/null +++ b/source/Physics/normalisevector.rst @@ -0,0 +1,52 @@ +.. _normalisinvector: + +Normalising a vector +==================== + +Normalising a vector is the process of making the length +of a vector equal to 1, while maintaining the direction. +This is called a `unit vector`. + +Formula +******* + +.. math:: + \overrightarrow{u} = { \overrightarrow{v} \over {\vert\vert\overrightarrow{v} \vert\vert} } + + +Intuition +********* + +Building an understanding of this process. +Starting out with a 2D vector :math:`\lt2, 4\gt`. + +.. image:: /images/normalisevectors1.png + +We can compute the hypotenuse with Pythagoras. + +:math:`\sqrt{2^2 + 4^2} = \sqrt{20}` + +So the question then becomes, "What can I do to :math:`\sqrt{20}` +to make it equal to 1". That's an easy question to answer, just +divide it by itself. **BUT** the catch is preserving the angle. + +Working out the angle we get :math:`\arcsin({4 \over \sqrt{20}}) \approx 63.43` + +Our `unit vector` will always be a length of 1. That means our vector point +will lie at 63.43 degrees on a unit circle. + +.. image:: /images/normalisevectors2.png + +To get our values we just plug in 63.43 into cosine and sine. + +.. math:: + \sin{(63.43)} = 0.89 + + \cos{(63.43)} = 0.45 + +Which is just our values of :math:`{4 \over \sqrt{20}}` and :math:`{2 \over \sqrt{20}}`. + +Putting it all together we can say + +.. math:: + {{\lt2,4\gt} \over \sqrt{20}} = \lt0.45, 0.89\gt diff --git a/source/images/normalisevectors1.png b/source/images/normalisevectors1.png new file mode 100644 index 0000000..d0342f4 Binary files /dev/null and b/source/images/normalisevectors1.png differ diff --git a/source/images/normalisevectors1.svg b/source/images/normalisevectors1.svg new file mode 100644 index 0000000..ae7fcb4 --- /dev/null +++ b/source/images/normalisevectors1.svg @@ -0,0 +1,167 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/source/images/normalisevectors2.png b/source/images/normalisevectors2.png new file mode 100644 index 0000000..b23c585 Binary files /dev/null and b/source/images/normalisevectors2.png differ diff --git a/source/index.rst b/source/index.rst index 573fb2a..1cf2d9f 100644 --- a/source/index.rst +++ b/source/index.rst @@ -5,3 +5,4 @@ Neo's Journal :glob: Mathematics/index.rst + Physics/index.rst