What laws do Probs. 1 and 4–7 illustrate?

Data from Prob. 1

a • b, b • a, b • c

Data from Prob. 4

|a + b|, |a| + |b|

Data from Prob. 5

Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].
Find:

|b + c|, |b| + |c|

Data from Prob. 6

Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].
Find:

|a + c|² + |a - c|² - 2(|a|² + |c|²)

Data from Prob. 7

Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].
Find:

|a • c|, |a||c|