Let OA = a, and PBB, then if PB be projected

on OA, we get P,B, and if we set up P1B, at 0 perpendicular to OA and move it parallel to itself along ОA, a rectangle is generated whose area is measured by

ОА.Р.В1.

Since the projection of PB is along OA, there is nothing to tell us in which plane P1B1 must be drawn. If P1B1 be revolved round OA, any one of the positions it takes up would do equally well. Hence the area determined by a and the projection of B on a must be considered as a scalar.

This may be looked at in a slightly different manner. Let B be the projection of ẞ on a as a vector, then ß, and a are like vectors which may be supposed to lie in the same line and cannot therefore determine any definite plane.

This product is written (aß), the first vector being the one on which the second is projected.

114. Definition. The Scalar Product (aß) of two vectors is the area of the rectangle contained by a and the projection of B on it, and is a scalar.

115.

Though (a) is a scalar, it may be positive or negative. If the projection B, of B on a has the sense

of a, the product is positive; if, as in the figure, it has the opposite sense, the product is negative.

From this we see that the algebraical rules for changing the sign of a product when one factor changes sign hold for the scalar product.

Two scalar products are equal if they are equal in magnitude and sign without regard to the position and direction of the vectors in space.

116. In the product (aß) if a and ẞ represent vector quantities, i.e. are not geometrical vectors, then the product is no longer an area. In particular if a is an ort, and B a line vector, then (aß) gives the length of the projection of B on a, i.e. (§ 113) the length of B1. While B1 itself, being a vector of length (aß) and direction a, is given by B1 = (aẞ) α.

117. Our two products can be expressed a little differently by the aid of Trigonometry. For if a and b denote the lengths of a and B and 0 be the angle between them, then the area of the parallelogram is ab sin 0, and if e be the ort giving the aspect of the area, then

These equations may be taken as the definitions of the sine and cosine of an angle, and trigonometrical formulae could be obtained from them. This branch of the Vector Calculus will be considered later.

118. For the vector product we have already seen that the Commutative Law does not hold, in fact

so that the Law is replaced by the following: A vector product changes sign on interchanging the two factors. For the scalar product however the Commutative Law does hold.

If we project ẞ on a (see figure) we get OB1, and

(aß) = OA. OB1.

or

and

But OA,A and OB,B are similar triangles, hence

since the product depends on the magnitude and sign only.

119. If either factor of a product of two vectors be multiplied by any scalar, then is the product multiplied by that scalar.

For if one side of a rectangle or parallelogram be increased m-fold then the area is increased m-fold. In symbols

and

[a.mẞ] = [ma. B] = m [aß]

(a.mB) = (ma. ẞ) = m (aß).

Hence if either factor of a product of two vectors has a scalar multiplier, the scalar may be taken outside the bracket.

120. The vector product of like vectors vanishes. Since like vectors are parallel, no area is swept out by moving one along the other, and the magnitude of the product is therefore zero.

Hence if is parallel to a, then [a] = 0.

The vector product differs therefore from an algebraical one in that it may vanish although neither factor vanishes. An instructive way of deducing this result is from the fact that

As special cases we may notice

(i) if neither a nor ẞ is zero, then a || B,

(ii) if [aß]=0 whatever a may be, then ß=0.

122. Scalar Product of Like Vectors. If B = a then for the scalar product (aß) we get (aa) which is the area of a square, the side of which represents the magnitude of a.

This scalar square is for shortness written a2.

If a be the length of a, then

(aa) = a2 = a2.

The length or magnitude of a vector a may be written Va? which it should be noted is not equal to a.

The ort in the direction of a can now be expressed in the form

α

123. If a is perpendicular to B, then ẞ has no projection on a and

(aẞ)=0.

The product may vanish therefore though neither factor is zero.

In the case of the rectangular orts 4 we have (44) = 42 = 1

These results should be compared with the corresponding ones for the vector product.

125. The distributive law holds for both products. The Scalar Product. The equation symbolising the Distributive Law may be put in two forms, which for the scalar product are really equivalent to one, since the Commutative Law holds.

The two forms are, for the scalar product,

(a|B+y)=(aß) + (ay)

(a + By) = (ay) + (By)

(i),

..(ii)

(the vertical within the first bracket is used to separate the two factors a and B+y, in order to avoid the introduction of further brackets).

then

In the figure let

OA = a, OB = B, BC = y,

OC = B + Y.