vault backup: 2026-05-24 20:13:08

.obsidian/plugins/harper/data.json

@@ -1,5 +1,5 @@
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+  "ignoredLints": "{\"context_hashes\":[214610641060740680,11253296314840191199,9548152354067699590,16721049695819994954,6119222255552956109,15865895689566000533,8216002975862858808]}",
   "useWebWorker": true,
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@@ -434,6 +434,7 @@
     "frc",
     "hea150",
     "math163",
+    "parallelpiped",
     "pcb"
   ],
   "dialect": 0,

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

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 
+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.