What is the purpose of cross multiplication in vector multiplication?

[hani14]

Hi,
what does it mean to cross multiply two vectors? I couldn't imagine them in real life.

eg Force vector.

Multiplying Force vector to a scalar value means you multiple the 'Strength' of the force,
Dot multiplication of Force with displacement to get work, means you get the work in direction of force.

but what about cross multiplication?

Thank you!

 

[mfb]

It leads to a vector orthogonal to both initial vectors, with a magnitude proportional to the magnitude of the individual vectors, and proportional to the sine of the angle between them.

This vector is important for charges moving in a magnetic field, for example. Why? Well, this is just the world we live in.

 

[jedishrfu]

As @mfb says the vector cross-product creates a vector that is orthogonal to both initial vectors and geometrically its magnitude is the area of the parallelogram made by the two initial vectors.

Given $C = A \times B$ then $|C| = |A| * |B| * sin(\theta)$

which you can interpret $|A|$ as the base of a parallelogram and $|B|sin(\theta)$ as the height of the parallelogram.

If $A \times B$ evaluates to zero then you can conclude that A is parallel to B ie the $sin(\theta) = 0$ meaning that the angle between them is 0 or 180 degrees.

Here's more on the vector cross product which is used extensively in physics and computer graphics:

https://en.wikipedia.org/wiki/Cross_product

It would be worth your while to understand its geometric properties and uses in science and math. There is a great set of videos on linear algebra that can strengthen your geometric understanding of the cross product and other vector / linear algebra principles: