43 lines
1.9 KiB
Markdown
43 lines
1.9 KiB
Markdown
#rs/class/math163 #math
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## Operations
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### Addition
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$$\vec{u} + \vec{v} = <u_{1} + v_{1}, u_{2} + v_{2}>$$
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### Subtraction
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$$\vec{u} - \vec{v} = <u_{1} - v_{1}, u_{2} - v_{2}>$$
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### Dot Product
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$$\vec{u} \cdot \vec{v} = <u_{1}v_{1}, u_{2}v_{2}>$$
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### Magnitude
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$$||\vec{u}|| = \sqrt{ u_{1}^2 + u_{2}^2}$$
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### Unit Vector
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$$ unit_u=\frac{\vec{u}}{||\vec{u}||}$$
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Used to find the vector of magnitude 1 in the same direction as $u$
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### Cross Product
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$$a \times b = \begin{vmatrix} i&j&k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}= i(a_2b_3-a_3b_2)-j(a_1b_3-a_3b_1)+k(a_1b_2-a_2b_1)$$
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Used to create a vector that is orthogonal to both $a$ and $b$. To find what direction it will point in use the right-hand rule (thumb, pointer and middle finger).
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The cross product is calculated by finding the [[Determinant of a Matrix|determinant]] of the matrix formed in the equation above with the unit vectors in the top row.
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Properties:
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- Not commutative $$u\times v \ne v \times u$$
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- Anti-commutative $$ u \times v = -(v \times u) $$
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- On self $$ a \times a = 0 $$
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The magnitude of the cross product of two vectors can be found with $$||u \times v|| = ||u|| * ||v|| * \sin(\theta)$$
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The magnitude of the cross product is equal to the area of the parallelogram formed with two adjacent sides as the vectors.
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### Projection
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$$proj_{u}v=\frac{u\cdot v}{\left|\left|u\right|\right|^{2}}u$$
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Used to project vector $v$ onto $u$
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![[VectorProjection.excalidraw]]
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### Triple Scalar Product
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Find the Determinant of:
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$$u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ v_{1} && v_{2} && v_{3} \\ w_{1} && w_{2} && w_{3} \end{vmatrix}$$
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The triple scalar product can also be found by finding the cross product of $v$ and $w$ and then taking the dot product of the resulting vector and $u$.
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$$u * (v \times w) = (u \times v) * w $$
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The triple scalar product is equal to the volume of the parallelpiped where each vector represents one adjacent edge. |