Mathematical Physics Vol 1

3.1 Vector algebra

63

Problem11

Give a geometric interpretation of the mixed product a · ( b × c ) .

Answer Consider a parallelepiped (Fig. 3.9) constructed over vectors a , b and c . Its volume is V = h · P , where h is the height, a P the area of the base. Given that the base is a parallelogram constructed over vectors b and c , according to the previous example it follows that P = | b × c | . Let n be a unit vector with direction orthogonal to the plane of the base (defined by vectors b and c ). The height of the parallelepiped, which corresponds to the base constructed over the vectors b and c , is equal to

( a · n = a · 1cos α , where α is the angle between vectors a and n ) ,

h = a · n

and thus it follows that

V = h · P = a · n | b × c | .

Given that

b × c = n | b × c |

we finally obtain

V = | a · ( b × c ) | .

(3.3)

Figure 3.9: Volume of the parallelepiped.

Thus the absolute value of the mixed product of three vectors is equal to the volume of the paral lelepiped formed by these vectors (the volume is equal to the absolute value, as it cannot be negative).

Problem12 Prove that a = b ⇔ a x = b x , a y = b y , a z = b z .

Problem13

Prove that

α a =[ α a x , α a y , α a z ] .