The scalar product of two vectors can be constructed by taking the component of one vector in the direction of the other and multiplying it times the magnitude of the other vector. This can be expressed in the form:
The magnitude of the vector product of two vectors can be constructed by taking the product of the magnitudes of the vectors times the sine of the angle (<180 degrees) between them. The magnitude of the vector product can be expressed in the form:
and the direction is given by the right hand rule.
F=ma is a scalar product, since, force is a vector quantity, m is scalar quantity,accelaration is vector quantity.
Some examples where scalar product is used,
The picture below gives its definition for two vectors a and b
The man is pulling the block with a constant force a so that it moves along the horizontal ground.The work done in moving the block through a distance b is then given by the distance moved through multiplied by the magnitude of the component of the force in the direction of motion.
This is |a||b|cost so we define the scalar or dot product as
a.b = |a||b| cos t
where t is the angle between a and b when they are placed tail to tail
To use the least amount of force possible, we would need to pull horizontally, so that we are pulling in the same direction as we want the object to move. Then we would have t=0 and cos t=1 so that
work done = a.b= magnitude of the force x distance moved in the direction of the force.
Now, for vector product example as to where you should use it:
If we have a force F acting through a point P with position vector r with respect to O, then F and r lie in a plane through O. I have also redrawn the force vector F shifted so that its tail is at O.
The torque or moment of F about an axis through O perpendicular to this plane is given by
T = r x F = |r||f| sin t n
the value of sin t does indeed give the various practical possibilities, when a person applies torque using a lever to turn on a tap.
Hope this helps.