Cross product
For two vectors in 3D, the cross product is a new vector at right angles to both. Its length is the area of the parallelogram they span.
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Analogy
The direction a screw moves when you turn it from a towards b.
Example
First: 2(6) - 3(5) = -3.Second: 3(4) - 1(6) = 6.Third: 1(5) - 2(4) = -3.Result (-3, 6, -3).
In practice
Gives normal directions to surfaces and torque in physics.
Practice
Flashcards
- What is the cross product of (1,2,3) and (4,5,6)?(-3, 6, -3).
Questions people ask
Is the cross product commutative?
No. Swapping the order reverses the direction: b cross a equals minus a cross b. Only the dot product is order-independent. This is why the cross product needs a careful right-hand rule to fix which way the new vector points.
Why does the cross product only work in 3D?
The standard cross product needs a unique direction perpendicular to two vectors. In 3D there is exactly one such line, but in 2D there is none, and in 4D there are many. Higher dimensions use other tools instead. Engineers mostly need it for forces, rotation and surfaces, which all live in ordinary three-dimensional space.