Exercise 1: Computing and Verifying a Cross Product — Possible Solution
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GIVEN
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e = [1, -2, 3]
f = [2, 0, -1]

STEP 1: COMPUTING e x f COMPONENT BY COMPONENT
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Using a x b = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1]:

  First component:  e2*f3 - e3*f2 = (-2)*(-1) - 3*0   = 2 - 0  = 2
  Second component: e3*f1 - e1*f3 = 3*2 - 1*(-1)       = 6 + 1 = 7
  Third component:  e1*f2 - e2*f1 = 1*0 - (-2)*2       = 0 + 4 = 4

  e x f = [2, 7, 4]

STEP 2: VERIFYING PERPENDICULARITY TO e
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(e x f) . e = (2)(1) + (7)(-2) + (4)(3) = 2 - 14 + 12 = 0

The result is 0, confirming e x f is perpendicular to e.

STEP 3: VERIFYING PERPENDICULARITY TO f
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(e x f) . f = (2)(2) + (7)(0) + (4)(-1) = 4 + 0 - 4 = 0

The result is 0, confirming e x f is also perpendicular to f.

WHY THIS WORKS AS AN ANSWER
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The cross product is computed directly from this chapter's own
component formula, showing each of the three subtraction pairs
separately, and the result is checked against this chapter's own
stated geometric property (perpendicular to both inputs) using the
dot product from Chapter 2 as the verification tool, rather than
simply trusting the formula was applied correctly.