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Dot Products And Cross Products

rename
Updated 2009-03-26 23:21

Dot Products and Cross Products

 

In the charts below, the following hold:
A and B are general vectors in cartesian x, y, z coordinates,
A = Ax x + Ay y + Az z,
B = Bx x + By y + Bz z,
x, y, and z are mutually-perpendicular unit vectors along the cartesian x, y, and z axes,
|A| is the magnitude or length of the vector A,
θ is the angle between the vectors A and B,
θ is 0 to 180 degrees or 0 to π radians,
AB is the dot product of the vectors A and B,
A × B is the cross product of the vectors A and B.

 

In the charts below, the term in the left column equals the term in the right column.

 

Left ColumnRight Column
|A|sqrt(Ax2 + Ay2 + Az2) ≥ 0
|B|sqrt(Bx2 + By2 + Bz2) ≥ 0
AA|A|2 ≥ 0
BB|B|2 ≥ 0
AAAx2 + Ay2 + Az2 ≥ 0
BBBx2 + By2 + Bz2 ≥ 0
ABAx Bx + Ay By + Az Bz
|x|=|y|=|z|1
xx = yy = zz1
xy = yx0
xz = zx0
yz = zy0
AB|A||B|cos(θ)
|A × B||A||B|sin(θ) ≥ 0
A × Ba vector perpendicular to both A and B
AB 0 if A and B are perpendicular
A × B 0 if A and B are parallel
x × yz
y × zx
z × xy
x × z-y
z × y-x
y × x-z
x × x = y × y = z × z0
memorize

 

Left ColumnRight Column
A × B   (Ay Bz - Az By) x
+ (Az Bx - Ax Bz) y
+ (Ax By - Ay Bx) z
B × A   (Az By - Ay Bz) x
+ (Ax Bz - Az Bx) y
+ (Ay Bx - Ax By) z
ABBA
A × B-(B × A)
A • (A × B)0
B • (A × B)0
|A × (A × B)||A|2 |B| sin(θ) ≥ 0
|B × (A × B)||B|2 |A| sin(θ) ≥ 0
memorize

 

In the next chart, the following hold:
a is a unit vector along A,
b is a unit vector along B,
A = Ab b + Aw w,
B = Ba a + Bv v,
v is a unit vector perpendicular to a and A so that Bv ≥ 0,
w is a unit vector perpendicular to b and B so that Aw ≥ 0.

 

Left ColumnRight Column
aA/|A|
bB/|B|
|A|sqrt(Ab2 + Aw2)
|B|sqrt(Ba2 + Bv2)
|A|2Ab2 + Aw2
|B|2Ba2 + Bv2
aa = bb = vv = ww1
a × a = b × b = v × v = w × w0
av = bw0
|a × v| = |b × w|1
abcos(θ)
|a × b| = aw = bvsin(θ) ≥ 0
Ab = Ab |A|cos(θ)
Aw = Aw|A|sin(θ) ≥ 0
Ba = Ba|B|cos(θ)
Bv = Bv|B|sin(θ) ≥ 0
ABAb |B| = Ba |A|
|A × B|Aw |B| = Bv |A| ≥ 0
B(AB)/(|B|2)Ab b
A(AB)/(|A|2)Ba a
A - B(AB)/(|B|2)Aw w
B - A(AB)/(|A|2)Bv v
memorize

 

In the next chart, C and D are two additional general vectors like A and B above:
C = Cx x + Cy y + Cz z,
D = Dx x + Dy y + Dz z.

 

Left ColumnRight Column
A • (B × C)   (Ax By Cz - Ax Bz Cy)
+ (Ay Bz Cx - Ay Bx Cz)
+ (Az Bx Cy - Az By Cx)
A • (B × C)  (Ax By Cz + Ay Bz Cx + Az Bx Cy)
- (Az By Cx + Ax Bz Cy + Ay Bx Cz)
B • (C × A)A • (B × C)
C • (A × B)A • (B × C)
(C × B) • A-A • (B × C)
(A × C) • B-B • (C × A)
(B × A) • C-C • (A × B)
A × (B × C)(AC)B - (AB)C
(A × B) • (C × D)(AC)(BD) - (AD)(BC)
memorize

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