Working With Vectors

WORKING WITH VECTORS

1.                  The direction of the vector V may be measured by an angle q from some known reference direction as shown in Fig. 1/1.

2.                  The negative of V is a vector -V having the same magnitude as V but directed in the sense opposite to V, as shown in Fig. 1/1.

  3.                  Vectors must obey the parallelogram law of combination. This law states that two vectors V₁ and V₂, treated as free vectors, Fig. 1/2a, may be replaced by their equivalent vector V, which is the diagonal of the parallelogram formed by V₁ and V₂ as its two sides, as shown in Fig. 1/2b. This combination is called the vector sum, and is represented by the vector equation

V = V₁ + V₂

Where the plus sign, when used with the vector quantities (in boldface type), means vector and not scalar addition. The scalar sum of the magnitudes of the two vectors is written in the usual way as V₁ + V₂. The geometry of the parallelogram shows that V ¹ V₁ + V₂.

4.                  Two vectors V₁ and V₂, again treated as free vectors, may also be added head-to-tail by the triangle law, as shown in Fig. 1/2c, to obtain the identical vector sum V. The order of addition of vectors does not affect their sum, so that V₁ + V₂ = V₂ + V₁. The difference V₁ - V₂ between the two vectors is easily obtained by adding -V₂ to V₁ as shown in Fig. 1/3, where either the triangle or parallelogram procedure may be used. The difference V¢ between the two vectors is expressed by the vector equation

V¢ = V₁ - V₂

Where the minus sign denotes vector subtraction.

 

5.                  Any two or more vectors whose sum equals a certain vector V are said to be the components of that vector. Thus, the vectors V₁ and V₂ in Fig. 1/4a are the components of V in the directions 1 and 2, respectively. It is usually most convenient to deal with vector components which are mutually perpendicular, these are called rectangular components. The vectors V_x and V_y in Fig. 1/4b are the x- and y-components, respectively, of V. Likewise, in Fig. 1/4c, V_x¢ and V_y¢ are the x¢- and y¢-components of V. When expressed in rectangular components, the direction of the vector with respect to, say, the x-axis is clearly specified by the angle q,

Where

6.                  A vector V may be expressed mathematically by multiplying its magnitude V by a vector n whose magnitude is one and whose direction coincides with that of V. The vector n is called a unit vector. Thus, In this way both the magnitude and direction of the vector are conveniently contained in one mathematical expression. In many problems, particularly three-dimensional ones, it is convenient to express the rectangular components of V, Fig. 1/5, in terms of unit vectors i, j, and k, which are vectors in the x-, y-, and z-directions, respectively, with unit magnitudes. Because the vector V is the vector sum of the components in the x-, y-, and z-directions, we can express V as follows:

We now make use of the direction cosines l, m, and n of V, which are defined by

Thus, we may write the magnitudes of the components of V as

Where, from the Pythagorean Theorem,

Note that this relation implies that