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ben committed 2026-05-24 20:13:08 -07:00
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.obsidian/plugins/harper/data.json
Vendored
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@@ -1,5 +1,5 @@
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"lintSettings": { "lintSettings": {
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@@ -434,6 +434,7 @@
"frc", "frc",
"hea150", "hea150",
"math163", "math163",
"parallelpiped",
"pcb" "pcb"
], ],
"dialect": 0, "dialect": 0,
Running Start/MATH&163 - Calculus 3/Class 10-9.md
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- triple scalar product: $$u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \end{vmatrix}$$ - triple scalar product: $$u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \end{vmatrix}$$
- (can either take the determinate of the matrix or the dot product of the cross product) - (can either take the determinate of the matrix or the dot product of the cross product)
- results in a scalar - results in a scalar
- This is equal to the volume of the parallelepiped with the three vectors representing adjacent edges - This is equal to the volume of the parallelpiped with the three vectors representing adjacent edges
- $$u * (v \times w) = (u \times v) * w $$ - $$u * (v \times w) = (u \times v) * w $$
- lines and planes in 3D space - lines and planes in 3D space
- can use 3D vectors to describe 3D lines - can use 3D vectors to describe 3D lines
Wiki/Math/Vector Basics.md
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@@ -15,4 +15,10 @@ Properties:
- Anti-commutative $$ u \times v = -(v \times u) $$ - Anti-commutative $$ u \times v = -(v \times u) $$
- On self $$ a \times a = 0 $$ - On self $$ a \times a = 0 $$
The magnitude of the cross product of two vectors can be found with $$||u \times v|| = ||u|| * ||v|| * \sin(\theta)$$ The magnitude of the cross product of two vectors can be found with $$||u \times v|| = ||u|| * ||v|| * \sin(\theta)$$
The magnitude of the cross product is equal to the The magnitude of the cross product is equal to the area of the parallelogram formed with two adjacent sides as the vectors.
### Triple Scalar Product
Find the Determinant of:
$$u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \end{vmatrix}$$
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 $u$.
$$u * (v \times w) = (u \times v) * w $$
The triple scalar product is equal to the volume of the parallelpiped where each vector represents one adjacent edge.