Geometric Interpretation of Vector Operations

It was mentioned earlier that a column or row vector with n elements (referred to hereafter as an n-vector) can be viewed as an n-tuple, and hence as a poini in an «-dimensional space (referred to hereafter as an w-space), Let us elaborate on this idea. In Fig. 4.2a7 a point (3,2) is plotted in a 2-space and is labeled «. This is the geometric counterpart of the vector or the vector u! — [ 3 2 ], both of which indicate in this context one and the u =

same ordered pair. If an arrow (a dirccted-linc segment) is drawn from the point of origin (0,0) to the point w, it will specify the unique straight route by which to reach the destination point u from the point of origin. Since a unique arrow exists for each point, we can regard the vector u as graphically represented either by the point (3, 2), or by the corresponding arrow. Such an arrow, which emanates from the origin (0, 0) like the hand of a clock, with a definite length and a definite direction, is called a radius vector.

1 The concept of scalar product is thus akin to the concept of inner product of two vectors with the same number of elements in each, which also yields a scalar. Recall; however, that the inner product is exempted from the conformability condition for multiplication, so that we may write it as u ■ v. In the case of scalar product (denoted without a dot between the two vector symbols), on the other hand, we can express it only as a row vector multiplied by a column vector, with the row vector in the lead.

Chapter 4 Linear Models and Matrix A Igebra 61

FIGURE 4.2

FIGURE 4.2

Chapter 4 Linear Models and Matrix A Igebra 61

(-3, -2)

Following this new interpretation of a vector, it becomes possible to give geometric meanings to (a) the scalar multiplication of a vector, (h) the addition and subtraction of vectors, and more generally, (c) the so-called linear combination of vectors.

First, if we plot the vector ^ = 2u in Fig. 4.2a, the resulting arrow will overlap the old one but will be twice as long. In fact, the multiplication of vector u by any scalar k will producc an overlapping arrow, but the arrowhead will be relocated, unless k — I. If the scalar multiplier is k > 1, the arrow will be extended out (scaled up); if 0 < k < I, the

; if k = 0, the arrow will shrink into the point of jj . A negative scalar multiplier will even reverse the direction of the arrow. Tf the vector a is multiplied by —1, for instance, we get —u —

^ , and this plots in Fig. 4.2b as an arrow of the same length as u but diametrically opposite in direction.

Next, consider the addition of two vectors, i; — \ and u =

arrow will be shortened (sealed down origin—which represents a null vector,

can be directly plotted as the broken arrow in Fig. 4.2c. If we construct a parallelogram with the two vectors u and v (solid arrows) as two of its sides, however, the diagonal of the parallelogram will turn out exactly to be the arrow representing the vector sum v + u. In general, a vector sum can be obtained geometrically from a parallelogram. Moreover, this method can also give us the vector difference v - u, sincc the latter is equivalent to the sum of v and (-l)w. In Fig. 4.2d, we first reproduce the vector v and the negative vector —u from diagrams c and A, respectively, and then construct a parallelogram. The resulting diagonal represents the vector difference v - u.

It lakes only a simple extension of these results to interpret geometrically a linear combination (i.e., a linear sum or difference) of vcctors. Consider the simple case of

The scalar multiplication aspect of this operation involves the relocation of the respective arrowheads of the two vcctors v and w, and the addition aspect calls for the construction ol a parallelogram. Beyond these two basic graphical operations, there is nothing new in u linear combination of vectors. This is true even if there are more terms in the linear combination. as in n

where kj are a set of scalars but the subscripted symbols vf now denote a set of vectors. To form this sum, the first two terms may be added first, and then the resulting sum is added to the third, and so forth, till all terms are included.

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