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Crossproducts and Equation Solving

In 3D statics, the equillibrium equations frequently contain expressions of the form

r F d * (†

r

some radius vector, length units or symbolic

F

some unknown force, symbolic

d

unit vector, usually obtained by dividing a distance vector by it's length

Let's try to do that symbolically

r.A a 2 a * 3 a * - 31 mat :

r.B 2 a * 2 - (a * a 31 mat :

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 :

r.A' a.0 2 a.0 * 3 a.0 * - 31 mat :

r.B' 2 a.0 * 2 - (a.0 * a.0 31 mat :

d.AB r.B r.A - : a 4 a * - 4 a * 31 mat

l.AB d.AB 2^sqrt : 33 sqrt a 2^sqrt *

d.AB' r.B' r.A' - (: 1 a.0 * 4 a.0 * - 4 a.0 * 31 mat

l.AB' d.AB' 2^sqrt : 5.74 a.0 *

Left cpolumn SMath cross product operator

Right cpolumn custom cross product function

SMath symbolic definition of unit vector

e.AB d.AB l.AB / : a 33 sqrt / 4 a * 33 sqrt / - 4 a * 33 sqrt / 31 mat a 2^sqrt /

e.AB' d.AB' l.AB' / : 0.174 0.696 - 0.696 31 mat

Scalar factor is a symbolic variable

optimization symbolic r.A A e.AB * Cross a 2 A a 33 sqrt / 4 a * 33 sqrt / - 4 a * 33 sqrt / 31 mat * a 2^sqrt / 31 el * 3 A a 33 sqrt / 4 a * 33 sqrt / - 4 a * 33 sqrt / 31 mat * a 2^sqrt / 21 el * + (* 11 mat

optimization symbolic r.A' A e.AB' * († #

optimization symbolic r.A' A e.AB' * Cross a.0 2 A 133 sqrt / 433 sqrt / - 433 sqrt / 31 mat * (31 el * 3 A 133 sqrt / 433 sqrt / - 433 sqrt / 31 mat * (21 el * + (* 11 mat

optimization symbolic r.A A e.AB * († #

optimization none r.A A e.AB * († a 2 a * 3 a * - 31 mat A 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - (2^sqrt / * †

optimization none r.A A e.AB * Cross 2 a * A 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - (2^sqrt / * (31 el * 3 a * - (A 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - (2^sqrt / * (21 el * - 11 mat

optimization none r.A' A e.AB' * († a.0 2 a.0 * 3 a.0 * - 31 mat A 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - (2^sqrt / * †

optimization none r.A' A e.AB' * Cross 2 a.0 * A 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - (2^sqrt / * (31 el * 3 a.0 * - (A 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - (2^sqrt / * (21 el * - 11 mat

symbolic r.A A e.AB * Cross Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

symbolic r.A A e.AB * († Maxima #

optimization symbolic r.A' A e.AB' * († Maxima #

optimization symbolic r.A' A e.AB' * Cross Maxima 4 a.0 * A * 33 sqrt / - 7 a.0 * A * 33 sqrt / - 6 a.0 * A * 33 sqrt / - 31 mat

none r.A A e.AB * († Maxima a - 4 a * 4 a * - 31 mat A * a 2 a * 3 a * - 31 mat † 1 a abs / 14 a abs * / 14 a abs * / 31 mat *

none r.A A e.AB * Cross Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

optimization none r.A' A e.AB' * († Maxima a.0 2 a.0 * 3 a.0 * - 31 mat A † a.0 4 a.0 * - 4 a.0 * 31 mat * 1 a.0 abs / 14 a.0 abs * / 14 a.0 abs * / 31 mat *

optimization none r.A' A e.AB' * Cross Maxima 4 a.0 * A * 33 sqrt / - 7 a.0 * A * 33 sqrt / - 6 a.0 * A * 33 sqrt / - 31 mat

Scalar factor is a unit

optimization symbolic r.A A e.AB * († #

optimization symbolic r.A A e.AB * Cross a 2 A a * 33 sqrt / 4 a * A * 33 sqrt / - 4 a * A * 33 sqrt / 31 mat a 2^sqrt / 31 el * 3 A a * 33 sqrt / 4 a * A * 33 sqrt / - 4 a * A * 33 sqrt / 31 mat a 2^sqrt / 21 el * + (* 11 mat

optimization symbolic r.A' A e.AB' * († 4 a.0 * A * 33 sqrt / - 7 a.0 * A * 33 sqrt / - 6 A * a.0 * 33 sqrt / - 31 mat

optimization symbolic r.A' A e.AB' * Cross 391988033247807 a.0 * A * 562949953421312 / - 8042387705153260 a.0 * A * 6600000000000000 / - 6893475175845650 a.0 * A * 6600000000000000 / - 31 mat

lastError Vektorprodukt ist nur für Vektoren mit drei Komponenten zulässig.

optimization none r.A' A e.AB' * († a.0 2 a.0 * 3 a.0 * - 31 mat A 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - (2^sqrt / * †

optimization none r.A A e.AB * Cross 2 a * A 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - (2^sqrt / * (31 el * 3 a * - (A 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - (2^sqrt / * (21 el * - 11 mat

optimization none r.A' A e.AB' * Cross 2 a.0 * 281474976710656404237659286799 / A * * 3 a.0 * - (281474976710656404237659286799 / A * - (* - 3 a.0 * - (1 5.74456264653803 / A * * a.0 281474976710656404237659286799 / A * * - a.0 281474976710656404237659286799 / A * - (* 2 a.0 * 1 5.74456264653803 / A * * - 31 mat

optimization none r.A A e.AB * († a 2 a * 3 a * - 31 mat A 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - 2 a * 2 - (a * a 31 mat a 2 a * 3 a * - 31 mat - (2^sqrt / * †

symbolic r.A A e.AB * Cross Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

symbolic r.A A e.AB * († Maxima 29 A * a 2^* 4 a abs * / -

optimization symbolic r.A' A e.AB' * († Maxima 4 A * a.0 * 33 sqrt / - 7 A * a.0 * 33 sqrt / - 6 A * a.0 * 33 sqrt / - 31 mat

optimization symbolic r.A' A e.AB' * Cross Maxima 391988033247807 A * a.0 * 562949953421312 / - 402119385257663 A * a.0 * 330000000000000 / - 137869503516913 A * a.0 * 132000000000000 / - 31 mat

optimization none r.A' A e.AB' * († Maxima 4 A * a.0 * 33 sqrt / - 7 A * a.0 * 33 sqrt / - 6 A * a.0 * 33 sqrt / - 31 mat

none r.A A e.AB * († Maxima a - 4 a * 4 a * - 31 mat A * a 2 a * 3 a * - 31 mat † 1 a abs / 14 a abs * / 14 a abs * / 31 mat *

none r.A A e.AB * Cross Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

optimization none r.A' A e.AB' * Cross Maxima 391988033247807 A * a.0 * 562949953421312 / - 402119385257663 A * a.0 * 330000000000000 / - 137869503516913 A * a.0 * 132000000000000 / - 31 mat

Maxima serves as a sort of symbolical \[REGION[62179413d2af4015aa7eb819f31cde43]]\

# eval

e.AB d.AB l.AB / Maxima : a 33 sqrt a abs * / 4 a * 33 sqrt a abs * / - 4 a * 33 sqrt a abs * / 31 mat

Scalar factor is a symbolic variable

optimization symbolic r.A A e.AB * († #

optimization symbolic r.A A e.AB * Cross a 2 A a 33 sqrt a abs * / 4 a * 33 sqrt a abs * / - 4 a * 33 sqrt a abs * / 31 mat * (31 el * 3 A a 33 sqrt a abs * / 4 a * 33 sqrt a abs * / - 4 a * 33 sqrt a abs * / 31 mat * (21 el * + (* 11 mat

optimization symbolic r.A' A e.AB' * († #

optimization symbolic r.A' A e.AB' * Cross a.0 2 A 133 sqrt / 433 sqrt / - 433 sqrt / 31 mat * (31 el * 3 A 133 sqrt / 433 sqrt / - 433 sqrt / 31 mat * (21 el * + (* 11 mat

optimization none r.A' A e.AB' * († a.0 2 a.0 * 3 a.0 * - 31 mat A 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - (2^sqrt / * †

optimization none r.A' A e.AB' * Cross 2 a.0 * A 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - (2^sqrt / * (31 el * 3 a.0 * - (A 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - (2^sqrt / * (21 el * - 11 mat

optimization none r.A A e.AB * († a 2 a * 3 a * - 31 mat A a 33 sqrt a abs * / 4 a * 33 sqrt a abs * / - 4 a * 33 sqrt a abs * / 31 mat * †

optimization none r.A A e.AB * Cross 2 a * A a 33 sqrt a abs * / 4 a * 33 sqrt a abs * / - 4 a * 33 sqrt a abs * / 31 mat * (31 el * 3 a * - (A a 33 sqrt a abs * / 4 a * 33 sqrt a abs * / - 4 a * 33 sqrt a abs * / 31 mat * (21 el * - 11 mat

symbolic r.A A e.AB * († Maxima #

symbolic r.A A e.AB * Cross Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

optimization symbolic r.A' A e.AB' * († Maxima #

optimization symbolic r.A' A e.AB' * Cross Maxima 4 a.0 * A * 33 sqrt / - 7 a.0 * A * 33 sqrt / - 6 a.0 * A * 33 sqrt / - 31 mat

none r.A A e.AB * († Maxima a 33 sqrt a abs * / - 4 a * 33 sqrt a abs * / 4 a * 33 sqrt a abs * / - 31 mat A * a 2 a * 3 a * - 31 mat †

none r.A A e.AB * Cross Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

optimization none r.A' A e.AB' * († Maxima a.0 2 a.0 * 3 a.0 * - 31 mat A † a.0 4 a.0 * - 4 a.0 * 31 mat * 1 a.0 abs / 14 a.0 abs * / 14 a.0 abs * / 31 mat *

optimization none r.A' A e.AB' * Cross Maxima 4 a.0 * A * 33 sqrt / - 7 a.0 * A * 33 sqrt / - 6 a.0 * A * 33 sqrt / - 31 mat

Scalar factor is a unit

optimization symbolic r.A A e.AB * († 4 A * a 2^* 33 sqrt a abs * / - 7 A * a 2^* 33 sqrt a abs * / - 6 A * a 2^* 33 sqrt a abs * / - 31 mat

optimization symbolic r.A A e.AB * Cross 4 A * a 2^* 33 sqrt a abs * / - 7 A * a 2^* 33 sqrt a abs * / - 6 A * a 2^* 33 sqrt a abs * / - 31 mat

optimization symbolic r.A' A e.AB' * († 4 a.0 * A * 33 sqrt / - 7 a.0 * A * 33 sqrt / - 6 A * a.0 * 33 sqrt / - 31 mat

optimization symbolic r.A' A e.AB' * Cross 391988033247807 a.0 * A * 562949953421312 / - 8042387705153260 a.0 * A * 6600000000000000 / - 6893475175845650 a.0 * A * 6600000000000000 / - 31 mat

This is the only way to get an exact result without Maxima. There must not be undefined variables in the expression except custom units.

optimization none r.A A e.AB * († a 2 a * 3 a * - 31 mat A a 33 sqrt a abs * / 4 a * 33 sqrt a abs * / - 4 a * 33 sqrt a abs * / 31 mat * †

optimization none r.A A e.AB * Cross 2 a * 4 a * A * 33 sqrt a abs * / * 3 a * - (4 a * A * 33 sqrt a abs * / - (* - 3 a * - (a A * 33 sqrt a abs * / * a 4 a * A * 33 sqrt a abs * / * - a 4 a * A * 33 sqrt a abs * / - (* 2 a * a A * 33 sqrt a abs * / * - 31 mat

optimization none r.A' A e.AB' * († a.0 2 a.0 * 3 a.0 * - 31 mat A 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - 2 a.0 * 2 - (a.0 * a.0 31 mat a.0 2 a.0 * 3 a.0 * - 31 mat - (2^sqrt / * †

optimization none r.A' A e.AB' * Cross 2 a.0 * 281474976710656404237659286799 / A * * 3 a.0 * - (281474976710656404237659286799 / A * - (* - 3 a.0 * - (1 5.74456264653803 / A * * a.0 281474976710656404237659286799 / A * * - a.0 281474976710656404237659286799 / A * - (* 2 a.0 * 1 5.74456264653803 / A * * - 31 mat

symbolic r.A A e.AB * († Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

symbolic r.A A e.AB * Cross Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

optimization symbolic r.A' A e.AB' * († Maxima 4 A * a.0 * 33 sqrt / - 7 A * a.0 * 33 sqrt / - 6 A * a.0 * 33 sqrt / - 31 mat

optimization symbolic r.A' A e.AB' * Cross Maxima 391988033247807 A * a.0 * 562949953421312 / - 402119385257663 A * a.0 * 330000000000000 / - 137869503516913 A * a.0 * 132000000000000 / - 31 mat

none r.A A e.AB * († Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

none r.A A e.AB * Cross Maxima 4 A * a abs * 33 sqrt / - 7 A * a abs * 33 sqrt / - 6 A * a abs * 33 sqrt / - 31 mat

optimization none r.A' A e.AB' * († Maxima 4 A * a.0 * 33 sqrt / - 7 A * a.0 * 33 sqrt / - 6 A * a.0 * 33 sqrt / - 31 mat

optimization none r.A' A e.AB' * Cross Maxima 391988033247807 A * a.0 * 562949953421312 / - 402119385257663 A * a.0 * 330000000000000 / - 137869503516913 A * a.0 * 132000000000000 / - 31 mat

General test for scalars in cross products

sym A.1 a.1 a.2 a.3 31 mat * A.2 * (B.1 b.1 b.2 b.3 31 mat * B.2 * († #

sym A.1 a.1 a.2 a.3 31 mat * A.2 * B.1 b.1 b.2 b.3 31 mat * B.2 * Cross A.1 a.1 a.2 a.3 31 mat * A.2 * (21 el B.1 b.1 b.2 b.3 31 mat * B.2 * (31 el * A.1 a.1 a.2 a.3 31 mat * A.2 * (31 el B.1 b.1 b.2 b.3 31 mat * B.2 * (21 el * - 11 mat

sym A.1 a.1 a.2 a.3 31 mat * A.2 * (B.1 b.1 b.2 b.3 31 mat * B.2 * († Maxima A.1 B.2 * a.1 * b.1 * A.1 B.2 * a.2 * b.2 * A.1 B.2 * a.3 * b.3 * 31 mat A.2 * B.1 †

sym A.1 a.1 a.2 a.3 31 mat * A.2 * B.1 b.1 b.2 b.3 31 mat * B.2 * Cross Maxima A.1 A.2 * B.1 * B.2 * a.2 b.3 * a.3 b.2 * - (* A.1 A.2 * B.1 * B.2 * a.3 b.1 * a.1 b.3 * - (* A.1 A.2 * B.1 * B.2 * a.1 b.2 * a.2 b.1 * - (* 31 mat

sym A.1 a.1 a.2 a.3 31 mat * A.2 * (B.1 b.1 b.2 b.3 31 mat * B.2 * († A.1 A.2 * B.1 * B.2 * a.2 b.3 * a.3 b.2 * - (* A.1 A.2 * B.1 * B.2 * a.3 b.1 * a.1 b.3 * - (* A.1 A.2 * B.1 * B.2 * a.1 b.2 * a.2 b.1 * - (* 31 mat

sym A.1 a.1 a.2 a.3 31 mat * A.2 * B.1 b.1 b.2 b.3 31 mat * B.2 * Cross A.1 A.2 * B.1 * B.2 * a.2 b.3 * a.3 b.2 * - (* A.1 A.2 * B.1 * B.2 * a.3 b.1 * a.1 b.3 * - (* A.1 A.2 * B.1 * B.2 * a.1 b.2 * a.2 b.1 * - (* 31 mat