1. Use the fact that i and j are orthogonal to determine:
   1.   i × j.   Hint: think of the right hand rule.
   2.  So what is   j × i ?

2. Use the same reasoning to determine:
   1.   j × k,
   2.   k × j,
   3.   k × i,
   4.   i × k,
   5.   i × i,
   6.   j × j,
   7.   k × k.

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Notes:

1. The triad i → j → k forms a right-handed basis for R³. (See Figure 17(b) on page 796.)
   Furthermore, any cyclical arrangement also forms a right-handed basis for R³, i.e.,
                j → k → i        and        k → i → j.
   Therefore,
   •   i × j = k,
   •   j × k = i,
   •   k × i = j.

2. Noncyclical arrangements do NOT form a right-handed basis for R³, i.e.,
                i → k → j,        k → j → i,        and        j → i → k.
   Therefore,
   •   j × i = –k,
   •   k × j = –i,
   •   i × k = –j.

3. Recall that the cross product of parallel vectors is always the zero vector 0. As a consequence:
   •   i × i = 0,
   •   j × j = 0,
   •   k × k = 0.

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Learn the properties in Theorem 11.

Here's an example showing how to use the triple scalar product to determine the volume of a parallelopiped.

Here's a summary of the cross product and vector properties.

DO THE HOMEWORK ON WEBASSIGN AFTER I COMPLETE THE SECTION.

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