cross product algebra and applications

(Example)

Cross Product Algebra and Applications

u × (v + w ) = u × v + u × w,

(au ) × (bv ) = ab(u × v).

Figure 17a, modernized: geometric interpretation behind bilinearity of the cross product.

For a right-handed Cartesian basis,

ˆi × ˆj = ˆk, ˆj × ˆk = ˆi, ˆk × ˆi = ˆj,

with reversal of order changing the sign.

Figure 17b, modernized: cyclic order of the right-handed Cartesian basis.

For components,

|--------||-----------||-| | | ˆi ˆj ˆk | | |u × v = ||ux uy uz ||.| (1) | |vx vy vz| | -----------------------|

Thus

u × v = (u v − u v )ˆi + (u v − u v )ˆj + (u v − u v )ˆk. y z z y z x x z x y y x

Example 1

For u = (2,−3, 5) and v = (−1, 4, 2),

u × v = (− 26, − 9, 5).

The result is orthogonal to both inputs, providing a direct check.

Example 2: plane through three points

For A = (1, 0, 2), B = (2, 3, 0), C = (−1,−2, 5),

− → −→ AB = (1,3,− 2), AC = (− 2,− 2,3),

so a Normal is

−→ −→ n = AB × AC = (5,1,4 ).

Hence the plane is

5(x − 1) + y + 4(z − 2) = 0,

|----------------------| -5x-+-y-+-4z-−-13-=-0.-|

Example 3: shortest distance between skew lines

If line directions are d₁,d₂, then n = d₁ × d₂ is normal to both. For points A on the first line and C on the second,

|---------------------| | −→ | | |AC--⋅ (d1-×-d2-)|| d = ∥d1 × d2∥ . | (2) -----------------------

Brand’s numerical example uses A = (2,−3, 1), B = (1, 0,−2), C = (4, 2, 1), D = (−1,−2, 1) and gives

51 d = √-----≈ 1.889. 730

Source problems

Find the plane through A = (1, 0, 2), B = (2, 3, 0), C = (−1,−2, 5).
Find a vector of length 39 perpendicular to (4,−3, 0) and (−4, 6, 1).
Find the plane through A = (1, 2, 3), B = (2,−3, 4) perpendicular to 3x+2y−z+7 = 0.
Find the plane through A = (−1, 2, 1) perpendicular to both x + 3y − 2z + 7 = 0 and 3x − 2y − 5z + 6 = 0.
Find the shortest distance from P = (1, 1,−3) to the plane through A = (1,−1, 2), B = (3, 2, 4), C = (6, 2,−2).
Find the shortest distance between lines AB and CD for:
1. A = (−2, 4, 3), B = (2,−8, 0), C = (1,−3, 5), D = (4, 1,−7);
2. A = (2, 3, 1), B = (0,−1, 2), C = (1, 2, 5), D = (−3, 1, 0).

Modern notation references

The notation and terminology in this modernized article follow standard present-day mechanics and vector-analysis usage, particularly:

J. R. Taylor, Classical Mechanics, University Science Books, 2005.
D. Kleppner and R. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison–Wesley, 2002.

Source

This article is a modernized restatement of the corresponding article in Louis Brand, Vectorial Mechanics, John Wiley & Sons, New York, 1930, Chapter I, “Vector Algebra.” The original 1930 edition is the source basis.

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cross product algebra and applications