Cross Products

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The cross product is the other famous way to multiply vectors. It only really lives in 3D, and instead of giving you a number like the dot product, it hands you back a brand new vector. It has a clean geometric meaning that makes it easy to remember.

Warm-up in 2D: signed area

Before 3D, picture two vectors v and w in the flat plane. They form a parallelogram. The 2D cross product is just the area of that parallelogram, with a sign attached.

v cross w = area of the parallelogram made by v and w

And here is the lovely part: that area is exactly the determinant you met in Chapter 6.

v = [a, c], w = [b, d]

v cross w = det | a  b | = ad - bc
                | c  d |

The sign tells you orientation. If w is counterclockwise from v, the result is positive. If it is clockwise, negative. So the 2D cross product is really "signed area", and you already know how to compute it.

The real cross product is in 3D

In 3D, the cross product of two vectors v and w is itself a 3D vector. It has two properties worth burning into memory.

1. Direction: the result points straight out, perpendicular to both v and w. It stands up off the plane that v and w lie in.
2. Length: the length of the result equals the area of the parallelogram formed by v and w.

length of (v cross w) = area of parallelogram of v and w

So a bigger or more spread-out parallelogram gives a longer result vector. If v and w point the same way, their parallelogram is flat with zero area, and the cross product is the zero vector.

Which of the two perpendicular ways does it point?

A line perpendicular to the plane has two directions: up or down. The cross product picks one using the right hand rule. Point your right hand's fingers along v, curl them toward w, and your thumb points the way v cross w goes.

This is why order matters and flips the sign:

w cross v = -(v cross w)

Swapping the inputs points the result the opposite way. Curl your fingers the other direction and your thumb flips.

How to compute it

For v = [v1, v2, v3] and w = [w1, w2, w3], the formula is:

v cross w = | v2*w3 - v3*w2 |
            | v3*w1 - v1*w3 |
            | v1*w2 - v2*w1 |

A worked example with v = [1, 0, 0] and w = [0, 1, 0] (that is î and ĵ):

v cross w = | 0*0 - 0*1 |   | 0 |
            | 0*0 - 1*0 | = | 0 |
            | 1*1 - 0*0 |   | 1 |

The answer is [0, 0, 1], which is k̂, pointing straight out of the page. That matches the picture: î and ĵ lie flat in the plane, and their cross product stands up perpendicular to both. The right hand rule confirms the direction.

Why this matters later

• Physics uses cross products everywhere: torque, angular momentum, and magnetic force are all "this vector crossed with that one".
• 3D graphics finds the direction a surface faces (its normal) by crossing two edges of a triangle. That normal decides how light bounces off it.
• Geometry uses the length of the cross product to get areas of triangles and parallelograms in space.

Quick gotchas

Dot product gives a number; cross product gives a vector. Do not mix them up.

Order matters. v cross w and w cross v point opposite ways.

Parallel vectors give zero. If two vectors point the same direction, their parallelogram is flat and the cross product is the zero vector.

True cross product is a 3D thing. The 2D version is really just the signed area (a number), not a vector.

What you walked away with

• The 3D cross product returns a vector perpendicular to both inputs.
• Its length is the area of the parallelogram the two vectors make.
• The right hand rule picks the direction, and swapping the order flips the sign.
• In 2D, the cross product is the signed area, which equals the determinant.

Next up, Chapter 11: a deeper look at why the cross product formula works, using the duality idea from Chapter 9. It is an optional, mind-bending bonus that ties several chapters together.