2.8 : Scalar Notation

Scalar notation is a useful method for simplifying calculations involving vectors. When vectors are added or subtracted, their components can be added or subtracted separately using scalar notation. For instance, force, a vector quantity, can be broken down into its x and y components, called rectangular components, and then the magnitude and direction of these components can be determined using trigonometric functions.

Consider a man pulling a rope from a hook in the northeast direction. The magnitude of this applied force vector is denoted as F₁. It is resolved into scalar components, represented as F_1x along the x-axis and F_1y along the y-axis. The expressions for the rectangular components F_1x and F_1y are obtained using trigonometric functions, as they form a right-angle triangle. Using these components and the Pythagorean theorem, the magnitude of the force F₁ can be calculated. The tan inverse of the y component over the x component gives the direction of the force. If another force F₂ acts on the same hook from the southeast direction, using a similar method, one can find the magnitude and direction of this force as well.

The resultant force is the algebraic sum of the components of both the forces along the x and y axes. Its magnitude can also be obtained by using the square root of the sum of the squares of its components. This resultant force can either represent the net force on an object or the force required to counteract the other forces.

Scalar notation is useful for calculating forces in different directions and understanding the forces acting on an object. By breaking forces into their rectangular components and then using trigonometric functions, one can determine the magnitude of force and the direction quickly and accurately.