bullet

Introduction to Vectors

In physics, in addition to the units associated to the quantities, all physical quantities are categorized in two groups:

Physics Quantities

Scalar quantities are defined only by their magnitude. The magnitude is the numerical value of the quantity; example, the time of a class period is 50 minutes. The magnitude is the number 50. The unit is the minutes. Remember that except for the case of counting objects, units are necessary for physics quantities. Scalar quantities are usually represented by italic type, t.

Time, Mass
Distance, Speed

Vector quantities are defined in terms of their magnitude and direction. The magnitude of a vector is the numerical value that defines the vector just as in the case of scalar quantities. The direction of a vector is as important as the magnitude when stating the characteristics of a vector. The direction of a vector on the surface of the Earth is usually defined in terms of the cardinal points (north, south, east, and west) or the xy-plane. Vertical directions are defined as up and down. Three dimensional vectors are represented in the regular three dimensional coordinate system. Vector quantities are represented by bold type, x, and/or by .

Weight
Displacement, Velocity

In one dimension, the difference between vector quantities and scalar quantities are minimal. The only difference is associated with the sign of the quantity.  The negative sign provides the direction for the vector. Here, the convention used is that vectors pointing to the right are positive while vectors pointing to the left are negative. This convention will be used along these notes.

Graphically, vectors can be represented by arrows such that the lengths of the arrows are proportional to the magnitudes of the vectors and the directions of the arrows to indicate the direction of the vectors. The head of the arrow is also called the tip while the bottom of the arrow is called the tail.

 

bullet

Operations with Vectors

Vectors as mathematical objects can be added, subtracted, and multiplied by scalars as well as by vectors. Here, the previous operations will be study from the point of view of their physical applications.

In the sequence of drawing at the left, the women originally walk three steps toward the right; then, she advanced still in the same direction two additional steps. At this point, she is at five steps from her starting point which is just the addition of the 3 steps + 2 steps. Therefore, based on observations such as the one just described, to add vectors that are pointing in the same direction, the magnitude of the resulting vector is obtained just from the result of adding signed numbers, and the direction of the resulting vector is the same as the direction of the original vectors. Notice that this result can be obtained following a graphical addition of these two vectors by placing the tail of the second vector at the tip of the first one, and the result is a vector starting at the tail of the first vector and ending at the tip of the second vector.

In these drawings, the woman first walk three steps straight in the direction due East, and then four additional steps in the direction due North. The result is that she walked from point A to point B. Alternatively she could walk straight from point A to point B. In this case, she will only need 5 steps in the direction straight from A to B (see below). With respect to the eastward direction of the drawing, the A to B direction can be calculated directly using the trigonometric inverse function of the tangent. The definition of the tangent is

which apply to the case of the drawing below results in

 

The number of steps can be calculated using the Pythagorean theorem for right angle triangles where the three steps East and the four steps North represent the sides of the triangle while the hypotenuse is the straight line from A to B (the hypotenuse is the longest side of the right angle triangle, it is also opposite to the right angle). Thus,

Therefore, the addition of vectors that are perpendicular to each other can be accomplish by employing the Pythagorean theorem for obtaining the magnitude of the resultant vector and the inverse of the trigonometric function tangent for obtaining its direction. In this case, it is also possible to obtain the result of the vector addition using the graphical method. The graphical method can also be used for adding vectors that are in any arbitrary direction as  demonstrated in the applet below.

  1. On the drawing, place the first vector maintaining the magnitude (to scale) and direction (with respect to the positive x axis) of the original vector, (advance applet)

  2. On the tip of the first vector, draw the second vector maintaining the magnitude and direction (accordingly with the horizontal axis), (advance applet)

  3. The resultant is the vector that starts at the tail of the first vector and end on the tip of the second vector, (advance applet)

  4. The characteristics of the resultant vector are the magnitude and the direction of the vector as measured from the horizontal line, (advance applet).

 

 

If vector R is the result of the vector addition of vectors A and B, R = A + B, and the magnitude of vector R is given by R = A -- B where A and  B are the magnitudes of vectors A and B respectively; then, it can be concluded that

 

a)

Vectors A and B are in the same direction.

 

N

b)

Vectors A and B are in opposite direction.

 

c)

Vectors A and B are perpendicular to each other

 

d)

Vectors A and B are 450 from each other

 

e)

None of the above.

 

 

bullet

 Components of Vectors (Two Dimensions)

In two dimensions, any vector can be represented as the vector addition of two perpendicular vectors, one along the x-axis and the other along the y-axis.

 

 

bullet
by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com