When to use inner product (scalar product, dot product) and outer product (cross product) when multiplying two vectors

When to use inner product (scalar product, dot product) and outer product (cross product) when multiplying two vectors

This problem is equivalent to when to add and subtract two quantities. Point multiplication and cross multiplication are two different operations. Using point multiplication or cross multiplication depends on what you want to calculate. For example, v = ω × R (linear velocity, angular velocity relationship), from physics knowledge, this multiplication is cross multiplication; w = f × R, from physics knowledge, this multiplication is point multiplication

Please explain the "vector product property"
Is vector multiplication itself equal to zero

Also known as "vector product."
For two vectors A.B, the same vectors (the same size and direction) are introduced from the origin o
OA.OB Then we make a vector OC from O, which is perpendicular to the plane OAB, and its length is equal to the triangle
The OAB area is 2 times of the value, and the direction is determined as follows:
When the thumb of the right hand points to OA and the index finger points to ob, OC passes through the palm of the hand vertically and upward
OC is denoted as OC = a × B
When the thumb of the right hand points to ob and the index finger points to OA, OC passes through the palm of the hand vertically and goes down
OC is denoted as OC = B × a

Shouldn't the product of vectors be scalar? But the formula of Lorentz force F = bqv, how can the product of two vectors and a scalar still be a vector

High school physics does not require this vector formula to write f = bqv on the line
In college physics, f = bqv is multiplied by a unit vector to express the direction, so B and V are vectors, bqv is scalar, and then f is vector by multiplying a unit vector, which represents the direction of Lorentz force