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FP3 Revision Thread

I'll kick off a P6 thread. Feel free to add your own notes.

Determinants.

2x2 Matrix A

a b
c d

|A| = ad - bc

3x3 Matrix B

a b c
d e f
g h i

|B| = a(ei - fh) - b(di - fg) + c(dh - eg)

|B| = 0 => matrix is inversible or points are coplanar or directions are linearly dependant.

Dot/Cross/Triple-Scalar Products

Dot Product
a.b = abcos@

(a,b,c).(x,y,z) = ax + by + cz
a.b = 0 => directions are perpendicular

Cross Product
axb = absin@
= area of parallelogram with sides a, b

(a,b,c)x(x,y,z) =
| i j k |
| a b c |
| x y z |
= i(bz - cy) - j(az - cx) + k(ay - bx)

Triple Scalar Product
a.(bxc) =
| a1 a2 a3 |
| b1 b2 b3 |
| c1 c2 c3 |

triple scalar product gives the area of a parallelepiped, (if it equals 0 it shows the points are co-planar)

Linear Combinations

(%,$,£)(A) = (p)

%(a) + $(b) + £(c) = p1
%(d) + $(e) + £(f) = p2
%(g) + $(h) + £(i) = p3

solve for the ratios % :colondollar:

:£ to find the combinations of the vectors to get the point.

Matrix Multiplication

A=
a b c
d e f
g h i

B =
j k l
m n o
p q r

AB =
(aj + bm + cp) (ak + bn + cq) (al + bo + cr)
(dj + em + fp) (dk + en + fq) (dl + eo + fr)
(gj + hm + ip) (gk + hn + iq) (dl + ho + ir)

BA =

(ja + kd + lg) (jb + ke + lh) (jc + kf + li)
(ma + nd + og) (mb + ne + oh) (mc + nf + oi)
(pa + qd + rg) (pb + qe + rh) (pc + qf + ri)