Two different methods exist for multipying vectors,
One method is associated with energy and work - a scalar - the SCALARor DOT product,
The other with vector resultants, the VECTOR or CROSS product.
SCALAR PRODUCT ( Bold = vectors, nonbold = sizes )
A . B = ABcosθ where θ = angle between them. Archetype is Work Done = Energy = F . s = Fscosθ
eg A billy-cart is pulled with a force of 30 N at an angle of 400 to the direction it makes, 47m along a road. What work does the little boy do on the cart?
Work done on the cart, probably against frictional forces = Fs cosθ = 30 x 47 x cos 400 joule = 1080 J
VECTOR PRODUCT
A X B = ABsinθ in size BUT the direction of the resultant is given by the right hand screw rule.
Archetype of this product are the magnetic force laws where B is given.
F = Il X B
F = qv X B
Other examples include angular momentum L = r X p and torque T = r X F
Both products can be calculated directly from the definitions but theyare often "easier" in unit ( component ) vector formats.