VECTOR ALGEBRA
is the forward direction of a right-handed screw rotating in the same sense as the
body. The velocity of any point in the body with position vectorris then given
byv=ω×r.
Since the basis vectorsi,j,kare mutually perpendicular unit vectors, forminga right-handed set, their vector products are easily seen to be
i×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 two
general vectorsaandbis given in terms of their components with respect to the
basis seti,j,k,by
a×b=(aybz−azby)i+(azbx−axbz)j+(axby−aybx)k. (7.33)For the reader who is familiar with determinants (see chapter 8), we record that
this can also be written as
a×b=∣
∣
∣
∣
∣
∣ijk
ax ay az
bx by bz∣
∣
∣
∣
∣
∣.That the cross producta×bis perpendicular to bothaandbcan be verifiedin component form by forming its dot products with each of the two vectors and
showing that it is zero in both cases.
Find the areaAof the parallelogram with sidesa=i+2j+3kandb=4i+5j+6k.The vector producta×bis given in component form by
a×b=(2× 6 − 3 ×5)i+(3× 4 − 1 ×6)j+(1× 5 − 2 ×4)k
=− 3 i+6j− 3 k.Thus the area of the parallelogram is
A=|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 our
discussion to define products of three vectors. Again, there are two possibilities,
thescalar triple productand thevector triple product.