4d3e9433-3715-479d-83f6-ff105946b4c0 9 3 5 false true 0 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Summary

What the red cells have in common is that they use vectorization.Vectorization can actually work, but only if used in a definition, not if just displaying an expression.

If we don't use units, vectorization is required but we are on thin ice.

v.a m 2 m * 3 m * 31 mat :

v.b 2 m * 2 m * 1 m * 31 mat :

wrong results

results not handled by cross product

vectorization, just result display

no vectorization, just result display

vectorization, definition context

no vectorization, definition context

use of the approaches to the left in the context of a cross product.

a b Cross a 21 el b 31 el * a 31 el b 21 el * - a 31 el b 11 el * a 11 el b 31 el * - a 11 el b 21 el * a 21 el b 11 el * - 31 mat Maxima :

v e v v v * sqrt / :

Normalization with the intuitive formula. None of the options provide a pure vector.
Vectorization yields rubbish.

v.a e A * vectorize A 2 A * 4 sqrt / 3 A * 9 sqrt / 31 mat

v.a e A * 114 sqrt / 214 sqrt / 314 sqrt / 31 mat A *

F v.a e A * : 114 sqrt / 214 sqrt / 314 sqrt / 31 mat A *

F.v v.a e A * vectorize : A 2 A * 4 sqrt / 3 A * 9 sqrt / 31 mat

v.b v.a e A * vectorize † 2 m * A * 34 sqrt * 9 sqrt - (* 4 sqrt 9 sqrt * / m A * 6 - 9 sqrt + (* 9 sqrt / m A * 2 - 4 sqrt + (* 31 mat

v.b v.a e A * † #

v.b F.v † 2 m * A * 34 sqrt * 9 sqrt - (* 4 sqrt 9 sqrt * / m A * 6 - 9 sqrt + (* 9 sqrt / m A * 2 - 4 sqrt + (* 31 mat

v.b F † #

v.b F Cross 4 m * A * 14 sqrt / 5 m * A * 14 sqrt / - 2 m * A * 14 sqrt / 31 mat

v.b v.a e A * Cross 4 m * A * 14 sqrt / 5 m * A * 14 sqrt / - 2 m * A * 14 sqrt / 31 mat

Normalization with procedure (line block). Vectorization gives a good result if used in a definition.
Yet just displaying the expression gives a different (wrong) result. So how would one explain that to students? Should we store intermediate results wherever possible? Wherever needed? How to tell when it is needed?

v# e l# v# v# * sqrt : v# l# / 21 line :

v.a e A * vectorize A 2 A * 4 sqrt / 3 A * 9 sqrt / 31 mat

v.a e A * 114 sqrt / 214 sqrt / 314 sqrt / 31 mat A *

F.v v.a e A * vectorize : A 14 sqrt / 2 A * 14 sqrt / 3 A * 14 sqrt / 31 mat

F v.a e A * : 114 sqrt / 214 sqrt / 314 sqrt / 31 mat A *

v.b v.a e A * vectorize † 2 m * A * 34 sqrt * 9 sqrt - (* 4 sqrt 9 sqrt * / m A * 6 - 9 sqrt + (* 9 sqrt / m A * 2 - 4 sqrt + (* 31 mat

v.b v.a e A * † #

v.b F.v † 4 m * A * 14 sqrt / 5 A * m * 14 sqrt / - 2 m * A * 14 sqrt / 31 mat

v.b F † #

v.b F Cross 4 m * A * 14 sqrt / 5 m * A * 14 sqrt / - 2 m * A * 14 sqrt / 31 mat

v.b v.a e A * Cross 4 m * A * 14 sqrt / 5 m * A * 14 sqrt / - 2 m * A * 14 sqrt / 31 mat

v e v v v * sqrt / Maxima :

Normalization with the intuitive formula and simplification by Maxima.
Vectorization is dangerous for expression display, yet required for a pure vector result
Non-vectorized expressions don't yield pure vectors.

v.a e A * vectorize A A A 31 mat

v.a e A * 114 sqrt / 214 sqrt / 314 sqrt / 31 mat A *

F.v v.a e A * vectorize : A 14 sqrt / 2 A * 14 sqrt / 3 A * 14 sqrt / 31 mat

F v.a e A * : 114 sqrt / 214 sqrt / 314 sqrt / 31 mat A *

v.b v.a e A * vectorize † m A * m A * - 031 mat

v.b v.a e A * † #

v.b F.v † 4 m * A * 14 sqrt / 5 A * m * 14 sqrt / - 2 m * A * 14 sqrt / 31 mat

v.b F † #

v.b F Cross 4 m * A * 14 sqrt / 5 m * A * 14 sqrt / - 2 m * A * 14 sqrt / 31 mat

v.b v.a e A * Cross 4 m * A * 14 sqrt / 5 m * A * 14 sqrt / - 2 m * A * 14 sqrt / 31 mat

v e v v norme / :

Normalization with vector norm function
Vectorization warrant a wrong result, \[REGION[7a0f0ed0401d4479aad164b62a3be7b4]]\ requires numeric evaluation

# norme

v.a e A * vectorize m A * m norme / 2 m * A * 2 m * norme / 3 m * A * 3 m * norme / 31 mat

v.a e A * A 0.267 0.535 0.802 31 mat

F.v v.a e A * vectorize : m A * m norme / 2 m * A * 2 m * norme / 3 m * A * 3 m * norme / 31 mat

F v.a e A * : A 0.267 0.535 0.802 31 mat

v.b v.a e A * vectorize † #

v.b v.a e A * † #

v.b F.v † #

v.b F † #

Normalization with the intuitive formula and the factor being a unit.
Vectorization now warrants rubbish
Non-vectorized expressions yield good result in both display and definition

v e v v v * sqrt / :

A A.0 :

v.a e A * vectorize A.0 2 A.0 * 4 sqrt / 3 A.0 * 9 sqrt / 31 mat

v.a e A * A.0 14 sqrt / 2 A.0 * 14 sqrt / 3 A.0 * 14 sqrt / 31 mat

F.v v.a e A * vectorize : A.0 2 A.0 * 4 sqrt / 3 A.0 * 9 sqrt / 31 mat

F v.a e A * : A.0 14 sqrt / 2 A.0 * 14 sqrt / 3 A.0 * 14 sqrt / 31 mat

v.b v.a e A * vectorize † 2 m * A.0 * 34 sqrt * 9 sqrt - (* 4 sqrt 9 sqrt * / m A.0 * 6 - 9 sqrt + (* 9 sqrt / m A.0 * 2 - 4 sqrt + (* 31 mat

v.b v.a e A * † 4 m * A.0 * 14 sqrt / 5 A.0 * m * 14 sqrt / - 2 m * A.0 * 14 sqrt / 31 mat

v.b F.v † 2 m * A.0 * 34 sqrt * 9 sqrt - (* 4 sqrt 9 sqrt * / m A.0 * 6 - 9 sqrt + (* 9 sqrt / m A.0 * 2 - 4 sqrt + (* 31 mat

v.b F † 4 m * A.0 * 14 sqrt / 5 A.0 * m * 14 sqrt / - 2 m * A.0 * 14 sqrt / 31 mat

Normalization with line procedure and the factor being a unit.
Vectorization gives rubbish unless used in a definition.

v# e l# v# v# * sqrt : v# l# / 21 line :

A A.0 :

v.a e A * vectorize A.0 2 A.0 * 4 sqrt / 3 A.0 * 9 sqrt / 31 mat

v.a e A * A.0 14 sqrt / 2 A.0 * 14 sqrt / 3 A.0 * 14 sqrt / 31 mat

F.v v.a e A * vectorize : A.0 14 sqrt / 2 A.0 * 14 sqrt / 3 A.0 * 14 sqrt / 31 mat

F v.a e A * : A.0 14 sqrt / 2 A.0 * 14 sqrt / 3 A.0 * 14 sqrt / 31 mat

v.b v.a e A * vectorize † 2 m * A.0 * 34 sqrt * 9 sqrt - (* 4 sqrt 9 sqrt * / m A.0 * 6 - 9 sqrt + (* 9 sqrt / m A.0 * 2 - 4 sqrt + (* 31 mat

v.b v.a e A * † 4 m * A.0 * 14 sqrt / 5 A.0 * m * 14 sqrt / - 2 m * A.0 * 14 sqrt / 31 mat

v.b F.v † 4 m * A.0 * 14 sqrt / 5 A.0 * m * 14 sqrt / - 2 m * A.0 * 14 sqrt / 31 mat

v.b F † 4 m * A.0 * 14 sqrt / 5 A.0 * m * 14 sqrt / - 2 m * A.0 * 14 sqrt / 31 mat

v e v v v * sqrt / Maxima :

Normalization with the intuitive formula and simplification by Maxima, factor is a unit.
Vectorization warrant a wrong result

A A.0 :

v.a e A * vectorize A.0 A.0 A.0 31 mat

v.a e A * A.0 14 sqrt / 2 A.0 * 14 sqrt / 3 A.0 * 14 sqrt / 31 mat

F.v v.a e A * vectorize : A.0 A.0 A.0 31 mat

F v.a e A * : A.0 14 sqrt / 2 A.0 * 14 sqrt / 3 A.0 * 14 sqrt / 31 mat

v.b v.a e A * vectorize † m A.0 * m A.0 * - 031 mat

v.b v.a e A * † 4 m * A.0 * 14 sqrt / 5 A.0 * m * 14 sqrt / - 2 m * A.0 * 14 sqrt / 31 mat

v.b F.v † m A.0 * m A.0 * - 031 mat

v.b F † 4 m * A.0 * 14 sqrt / 5 A.0 * m * 14 sqrt / - 2 m * A.0 * 14 sqrt / 31 mat

v e v v norme / :

Normalization with vector norm function and factor is a unit.
Vectorization warrant a wrong result, \[REGION[7a0f0ed0401d4479aad164b62a3be7b4]]\ requires numeric evaluation

# norme

A A.0 :

v.a e A * vectorize m A.0 * m norme / 2 m * A.0 * 2 m * norme / 3 m * A.0 * 3 m * norme / 31 mat

v.a e A * 0.267 A.0 * 0.535 A.0 * 0.802 A.0 * 31 mat

F.v v.a e A * vectorize : m A.0 * m norme / 2 m * A.0 * 2 m * norme / 3 m * A.0 * 3 m * norme / 31 mat

F v.a e A * : 0.267 A.0 * 0.535 A.0 * 0.802 A.0 * 31 mat

v.b v.a e A * vectorize † #

v.b v.a e A * † 1.07 m * A.0 * 1.34 m * A.0 * - 0.535 m * A.0 * 31 mat

v.b F.v † #

v.b F † 1.07 m * A.0 * 1.34 m * A.0 * - 0.535 m * A.0 * 31 mat