Dot Products And Cross Products
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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,
A • B 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 Column | Right Column |
|---|---|
| |A| | sqrt(Ax2 + Ay2 + Az2) ≥ 0 |
| |B| | sqrt(Bx2 + By2 + Bz2) ≥ 0 |
| A • A | |A|2 ≥ 0 |
| B • B | |B|2 ≥ 0 |
| A • A | Ax2 + Ay2 + Az2 ≥ 0 |
| B • B | Bx2 + By2 + Bz2 ≥ 0 |
| A • B | Ax Bx + Ay By + Az Bz |
| |x|=|y|=|z| | 1 |
| x • x = y • y = z • z | 1 |
| x • y = y • x | 0 |
| x • z = z • x | 0 |
| y • z = z • y | 0 |
| A • B | |A||B|cos(θ) |
| |A × B| | |A||B|sin(θ) ≥ 0 |
| A × B | a vector perpendicular to both A and B |
| A • B | 0 if A and B are perpendicular |
| A × B | 0 if A and B are parallel |
| x × y | z |
| y × z | x |
| z × x | y |
| x × z | -y |
| z × y | -x |
| y × x | -z |
| x × x = y × y = z × z | 0 |
| Left Column | Right 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 |
| A • B | B • A |
| 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 |
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 Column | Right Column |
|---|---|
| a | A/|A| |
| b | B/|B| |
| |A| | sqrt(Ab2 + Aw2) |
| |B| | sqrt(Ba2 + Bv2) |
| |A|2 | Ab2 + Aw2 |
| |B|2 | Ba2 + Bv2 |
| a • a = b • b = v • v = w • w | 1 |
| a × a = b × b = v × v = w × w | 0 |
| a • v = b • w | 0 |
| |a × v| = |b × w| | 1 |
| a • b | cos(θ) |
| |a × b| = a • w = b • v | sin(θ) ≥ 0 |
| Ab = A • b | |A|cos(θ) |
| Aw = A • w | |A|sin(θ) ≥ 0 |
| Ba = B • a | |B|cos(θ) |
| Bv = B • v | |B|sin(θ) ≥ 0 |
| A • B | Ab |B| = Ba |A| |
| |A × B| | Aw |B| = Bv |A| ≥ 0 |
| B(A • B)/(|B|2) | Ab b |
| A(A • B)/(|A|2) | Ba a |
| A - B(A • B)/(|B|2) | Aw w |
| B - A(A • B)/(|A|2) | Bv v |
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 Column | Right 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) | (A • C)B - (A • B)C |
| (A × B) • (C × D) | (A • C)(B • D) - (A • D)(B • C) |
