Examples for The vector product

Example 1

i × j = k

We calculate

| | |i j k| | | | | | | i × j = ||1 0 0||= ||0 0||i- ||1 0||j + ||1 0||k = k. || || |1 0| |0 0| |0 1| 0 1 0

Given u = i - 2j + 2k and v = 4i - 3k. Calculate

We use the formula for vector products in terms of components $| | ||i j k || u × v = |1 - 2 2 | = 6i + 11j + 8k. ||4 0 -3||$
The magnitude of u × v is 6² + (11)² + 8² = 221. On the other hand if we write for the angle between the vectors u, v it follows from the definition of vector product the magnitude of u × v is also given by the formula $|u × v|= |u ||v|sin h.$
We have therefore
$V~ 221 = |u × v| = |u||v |sin h = (3)(5)sinh,$
and so
$V~ sinh = --221. 15$