Final answer:
The cross product of vectors a and b can be found using the formula: a × b = (aybz - azby)i + (azbx - axbz)j + (axby - aybx)k. Substituting the values of a and b into the formula, we can find the cross product. To verify that the cross product is orthogonal to both a and b, we can calculate the dot products.
Step-by-step explanation:
The cross product of two vectors, denoted as a × b, is a vector that is perpendicular to both vectors a and b. To find the cross product, we can use the following formula:
a × b = (aybz - azby)i + (azbx - axbz)j + (axby - aybx)k
For the given vectors a = (2, 3, 0) and b = (1, 0, 9), substituting the values into the formula, we have:
a × b = (3*9 - 0*0)i + (0*1 - 2*9)j + (2*0 - 3*1)k
Simplifying the equation, we get:
a × b = 27i - 18j - 3k
To verify that the cross product is orthogonal to both a and b, we can calculate their dot products:
(a × b) · a = (27i - 18j - 3k) · (2i + 3j + 0k) = 54 - 54 + 0 = 0
(a × b) · b = (27i - 18j - 3k) · (1i + 0j + 9k) = 27 - 0 + 27 = 54