1A man cuts grass by pushing a lawnmower along its handle with a force of F N at an angle θ to the ground, and the mower moves a distance d. Among the force applied by the man, the displacement and the amount of work, which is a vector and which is a scalar? Why?
2How can the work done by the man be expressed as an equation?
3How much work has the man done?
4A girl pushes a trolley on a flat road with a force of F N in a straight line, and it moves with uniform velocity v m/s. Among power, force and velocity, which are vector quantities and which are scalar quantities? Give reasons.
5Write the equation showing the relation between power, force and velocity.
6How much power has the girl applied to the trolley?
7Define the scalar product of two vectors. What is its other name?
8Why is the product \( \vec{a}\cdot\vec{b} \) called a scalar product?
9If \( |\vec{a}| = 6 \), \( |\vec{b}| = 5 \) and the angle between them is \( 60^\circ \), find \( \vec{a}\cdot\vec{b} \).
10If \( |\vec{a}| = 4 \), \( |\vec{b}| = 3 \) and \( \vec{a}\cdot\vec{b} = 6\sqrt{3} \), find the angle between \( \vec{a} \) and \( \vec{b} \).
11A person drags a box 10 m along the floor with a rope, pulling with a force of 100 N at \( 30^\circ \) to the floor. Find the work done.
12Who invented quaternions, and who simplified the idea into modern vector analysis?