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Cross-training for cross products next chapter
Dot products measure alignment along a direction; cross products (next chapter) build perpendicular...
Geometric definition in $\mathbb{R}^3$
The cross product mathbfatimesmathbfb in mathbbR^3 is a vector perpendicular to both mathbfa and...
Bilinearity and algebraic determinant template
Cross product is bilinear: linear in each argument when the other is fixed. Distribution rules...
Orthogonality identities
By construction, mathbfacdot(mathbfatimesmathbfb)=0 and mathbfacdot(mathbfbtimesmathbfa)=0: the...
Connection to $2\times2$ determinant as area
Projecting to coordinate planes, cross magnitude recovers planar areas used in determinant videos...
Torque and angular momentum hooks
Mechanics uses boldsymboltau=mathbfrtimesmathbfF because only the perpendicular lever arm...
Cross products set up transformed versions next chapter
The next chapter asks how T(mathbfatimesmathbfb) relates to T(mathbfa)times T(mathbfb) for a linear...
The naive cross-interchange identity usually fails
For a general linear map T:mathbbR^3tomathbbR^3, do not assume T(mathbfatimesmathbfb)=Tmathbfatimes...
Adjugate / cofactor matrix viewpoint (conceptual)
The adjugate (classical adjugate) records how normals and oriented 2-faces transform under a linear...
Orthogonal maps split into rotations and reflections
Orthogonal maps satisfy Q^TQ=I. Determinant det Q=pm1 separates proper rotations (det Q=1) from...
Computing examples without memorizing a long formula
Practical strategy: track basis images Thatmathbfi,Thatmathbfj,Thatmathbfk and expand cross...
Edge cases with rank-deficient $T$
Kernel directions create ambiguity: many different inputs can share the same output under a...