Question 4: Can the scalar product of two vectors be negative? Provide a proof and give an example
Yes, the scalar product of two vectors can be negative.
Proof: Consider two forces A and B acting at an angle θ with one another. There scalar product is defined as,
This means the scalar product depends upon the magnitude of A, magnitude of B and the cosine value of the angle θ between them. If one of them is negative, the scalar product can be negative. Now the cosine value can be negative; for example cos180° is -1. Therefore, in case the angle between the two vectors is 180°, the product is negative.
Work done is scalar product of force and displacement. The classical example of negative scalar product (or negative work as in this case) is the work done against the force of friction. When a surface is dragged against the other, the movement is resisted by the force of friction between the two surfaces. The displacement thus occurred is the work done against this friction.
Now the displacement occurs in one direction and the friction is completely in opposite direction to it. Therefore, the angle between displacement and force of friction is 180° and the scalar product is negative.