Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR ALGEBRAis the forward direction of a right-handed screw rotating in the same sense as thebody. The velocity of any point in the body with position vector r is then givenby v = ω × r.Since the basis vectors i, j, k are mutually perpendicular unit vectors, forminga right-handed set, their vector products are easily seen to bei × i = j × j = k × k = 0, (7.29)i × j = −j × i = k, (7.30)j × k = −k × j = i, (7.31)k × i = −i × k = j. (7.32)Using these relations, it is straightforward to show that the vector product of twogeneral vectors a and b is given in terms of their components with respect to thebasis set i, j, k, bya × b =(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. (7.33)For the reader who is familiar with determinants (see chapter 8), we record thatthis can also be written as∣ i j k ∣∣∣∣∣a × b =a x a y a z .∣ b x b y b zThat the cross product a × b is perpendicular to both a and b can be verifiedin component form by forming its dot products with each of the two vectors andshowing that it is zero in both cases.◮Find the area A of the parallelogram with sides a = i +2j +3k and b =4i +5j +6k.The vector product a × b is given in component form bya × b =(2× 6 − 3 × 5)i +(3× 4 − 1 × 6)j +(1× 5 − 2 × 4)k= −3i +6j − 3k.Thus the area of the parallelogram isA = |a × b| = √ (−3) 2 +6 2 +(−3) 2 = √ 54. ◭7.6.3 Scalar triple productNow that we have defined the scalar and vector products, we can extend ourdiscussion to define products of three vectors. Again, there are two possibilities,the scalar triple product and the vector triple product.224