What are the important points when dealing with scalars and vectors? I have summarized down below some four important concepts you need to know when solving problems involving scalars and vectors.
- A scalar can be a negative or positive number. Scalar has magnitude but without direction.
- A vector is a quantity that has a magnitude, direction, and sense.
- When you multiply a vector by a negative scalar, the direction of the vector will change.
- If two collinear vectors are added together, the answer is just the algebraic (or scalar) addition of the magnitudes.
The next question you may have in mind is, “How do I add vectors?”
Adding two vectors can be done in two different ways–one is through the parallelogram law and the other one is the trigonometry.
The book of Russel Hibbeler has outlined the procedures in doing this.
THE PARALLELOGRAM LAW
- Two “component” forces and add according to the parallelogram law, yielding a resultant force that forms the diagonal of the parallelogram.
- If a force is to be resolved into components along two axes u and v, then start at the head of force and construct lines parallel to the axes, thereby forming the parallelogram. The sides of the parallelogram represent the components, and
- Label all the known and unknown force magnitudes and the angles on the sketch and identify the two unknowns as the magnitude and direction of , or the magnitudes of its components.
- Redraw a half portion of the parallelogram to illustrate the triangular head-to-tail addition of the components.
- From this triangle, the magnitude of the resultant force can be determined using the law of cosines, and its direction is determined from the law of sines. The magnitudes of two force components are determined from the law of sines.
LAW OF SINES
LAW OF COSINES
The formulas for the sine law and cosine law above use variables from the triangle shown below. Take note that the relationships between the sides and the angles are very crucial in using the formulas.