(^12) Vector Operations
Furthermore, if A is any real scalar, then (Xu) v = X(u-v). For example
(2u) • v = 2(ii • v). Also (—it) • v = — (u • v), since the angle between — u
and v is-K — 9 and cos(7r — 6) = — cos6. These observations are summarized
by the formula
(\u + (iv) • w = (u • w) + fi(v • w), (1.2.5)
where A and /U are any real scalars.
Note that these properties of inner product are independent of any co-
ordinate system.
Next, we will find an expression for the inner product in terms of the
components of the vectors in cartesian coordinates. Let i, j, « be an
orthonqrmal set forming a cartesian coordinate system. Any position vector
x = OP can be expressed as x = xii+X2j+X3K, where xi = xi, X2 = xj
and X3 = x • fi, are the cartesian coordinates of the point P. If y is another
vector, then y = yii+y2j+y3& and by (1.2.5) and the orthonormal property
of i, j, K, we get
x • y = xiy! + x 2 y2 + X3V3- (1.2.6)
This formula expresses xy in terms of the coordinates of a; and y. Together
with (1.2.3), we can use the result for computing the angle between two
vector with known components in a given cartesian coordinate system.
In linear algebra and in some software packages, such as MATLAB,
vectors are represented as column vectors, that is, as 3 x 1 matrices; for
a summary of linear algebra, see page 451. If x and y are column vectors,
then the transpose xT is a row vector (1x3 matrix) and, by the rules of
matrix multiplication, x • y = xTy.
EXERCISE 1.2.1.c Let x, y be column vectors and, A a 3 x 3 matrix. Show
that Axy = y^7 'Ax = xTATy = ATy • x. Hint: (AB)T = BTAT.
We can now summarize the main properties of the inner product:
(11) u • u > 0 and u • u = 0 if and only if u = 0.
(12) (Ait + /j,v) • w = {u • w) + n(v • w), where A, fj, are real numbers.
(13) uv = vu.
(14) u • v = 0 if and only if u _L v.
Property (14) includes the possibility u = 0 or v = 0, because, by conven-
tion, the zero vector 0 does not have a specific direction and is therefore
coco
(coco)
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