Yesterday Rayan asked an interesting question. " Mathematically speaking, why is work a scalar?" In other words, why does multiplying two vectors by the cosine of the angle between them give a scalar. Not easy to answer without representing the vectors by matrices but here's an attempt.

What is the difference between a vector and a scalar? Let´s consider the vector in the animation and the distance between the points A and C. When I flip the axis to view from the other side the vector has a different direction but the distance is the same.

Now let´s take two vectors.

uvcosθ will give a number but is it a vector or scalar?

If we flip the axis

The angle is now upside down so we get uvcos(-θ) which is the same as before since cosθ = cos(-θ). The result is therefore a scalar.

If we now consider uvsinθ we see that this changes sign when the axis flips since sinθ = -sin(-θ) the result is a vector. Change of sign implies opposite direction, the vector in question is pointing out of the page, when the axis is flipped it points in.