Mathematical Physics Vol 1

3.1 Vector algebra

57

b)

a · b = b · a .

c)

a · ( b + c )= a · b + a · c .

d)

α ( a · b )=( α a · b ) ,

where α is a real number.

Proof

a) For a = 0 it is obvious that a · a = 0. Let a̸ = 0, then at least one of the values a x , a y or a z is different from zero. Thus, the following is true a · a = | a | 2 = a 2 x + a 2 y + a 2 z > 0 . b) a · b = | a |·| b | cos θ = | b |·| a | cos ( − θ )= b · a

which proves the commutativity of the scalar product (Fig. 3.4 (a)). We have used here the fact cosine is an even function, namely cos θ = cos ( − θ )

Figure 3.3: Figure in Problem 4.

c) Let e a be the unit vector in the direction of vector a . The projection of vector ( b+c ) on vector a is equal to the sum of projections of vectors b and c on vector a (see Fig. 3.4 (b))

( b + c ) · e a = b · e a + c · e a   ·| a | ( b + c ) · e a ·| a | = b · e a ·| a | + c · e a ·| a | = =( b + c ) · a = b · a + c · a ⇒ a · ( b + c )= a · b + a · c .

Figure 3.4: Figure in Problem 4.

d)

α ( a · b )= α ( a · b cos θ )= α a · b cos θ =( α a ) · b cos θ =( α a · b ) .