diff --git a/.obsidian/plugins/harper/data.json b/.obsidian/plugins/harper/data.json index d70f516..e567095 100644 --- a/.obsidian/plugins/harper/data.json +++ b/.obsidian/plugins/harper/data.json @@ -1,5 +1,5 @@ { - "ignoredLints": "{\"context_hashes\":[15865895689566000533,6119222255552956109,11253296314840191199,16721049695819994954,8216002975862858808,214610641060740680,9548152354067699590]}", + "ignoredLints": "{\"context_hashes\":[214610641060740680,11253296314840191199,9548152354067699590,16721049695819994954,6119222255552956109,15865895689566000533,8216002975862858808]}", "useWebWorker": true, "lintSettings": { "ACoupleMore": null, @@ -434,6 +434,7 @@ "frc", "hea150", "math163", + "parallelpiped", "pcb" ], "dialect": 0, diff --git a/Running Start/MATH&163 - Calculus 3/Class 10-9.md b/Running Start/MATH&163 - Calculus 3/Class 10-9.md index 57d88a6..f886c27 100644 --- a/Running Start/MATH&163 - Calculus 3/Class 10-9.md +++ b/Running Start/MATH&163 - Calculus 3/Class 10-9.md @@ -7,7 +7,7 @@ - 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) - 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 - $$u * (v \times w) = (u \times v) * w $$ - lines and planes in 3D space - can use 3D vectors to describe 3D lines diff --git a/Wiki/Math/Vector Basics.md b/Wiki/Math/Vector Basics.md index d96014e..d0c0b23 100644 --- a/Wiki/Math/Vector Basics.md +++ b/Wiki/Math/Vector Basics.md @@ -15,4 +15,10 @@ Properties: - Anti-commutative $$ u \times v = -(v \times u) $$ - 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 is equal to the \ No newline at end of file +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. \ No newline at end of file