Mathematical Physics Vol 1

Chapter 3. Examples

56

b) Consider vectors a , b and c (Fig. 3.2). Let us first add vectors a and b ( OP + PQ = OQ ) , and then add vector c to the resulting vector (sum) ( OQ + QR = OR ) , thus obtaining vector d .

Let us now add first vectors b and c ( −→ PQ + −→ QR = −→ PR ) , and then add vector a to the resulting vector ( −→ OP + −→ PR = −→ OR . We have thus once again obtained the vector d , which proves the associativ ity of the vector sum. ( a + b )+ c = a +( b + c )= d .

Figure 3.2: Figure in Problem 2.

Problem3

Prove

i j k a x a y a z b x b y b z

a × b =

.

Proof

a × b =( a x i + a y j + a z k ) × ( b x i + b y j + b z k )= = a x b x i × i + a x b y i × j + a x b z i × k + a y b x j × i + a y b y j × j + a y b z j × k + + a z b x k × i + a z b y k × j + a z b z k × k = =( a y b z − a z b y ) i +( a z b x − a x b z ) j +( a x b y − a y b x ) k = = determinant .

R Note that this is a "symbolic" determinant. Namely, a determinant yields a scalar value, while in this case the result is a vector. However, the determinant properties are valid for this "determinant" as well.

Problem4

Prove: a)

2 > 0 , except for a = 0 .

a · a = | a |