Vectors - 3D Math Primer for Graphics and Game Development - page 74

13. Calculate a×b and b×a for the following vectors:

, b = 0 (a) a = 0 −1 0 0 1

, b = 1 −2 −1 (b) a = −2 4 1

(c) a = 3 10 7 , b = 8 −7 4

14. Prove the equation for the magnitude of the cross product

a×b = absin θ.

(Hint: make use of the geometric interpretation of the dot product and try
to show how the left and right sides of the equation are equivalent, rather
than trying to derive one side from the other.)
15. Section 2.8 introduced the norm of a vector, namely, a scalar value asso-
ciated with a given vector. However, the definition of the norm given in
that section is not the only definition of a norm for a vector. In general,
the p-norm of an n-dimensional vector is defined as

1/p

n

|x i | p x

p

≡ .

i=1

Some of the more common p-norms include:
• The L 1 norm, a.k.a. Taxicab norm (p = 1):

≡ n

x |x i |.

1

i=1

• The L 2 norm, a.k.a. Euclidean norm (p = 2). This is the most
common and familiar norm, since it measures geometric length:

n

x ≡ x i .

2

i=1

• The infinity norm, a.k.a. Chebyshev norm (p = ∞):

x ∞ ≡ max (|x 1 |, . . . ,|x n |) .

Each of these norms can be thought of as a way to assigning a length or size
to a vector. The Euclidean norm was discussed in Section 2.8. The Taxicab
norm gets its name from how a taxicab would measure distance driving the
streets of a city laid out in a grid (e.g., Cartesia from Section 1.2.1). For
example, a taxicab that drives 1 block east and 1 block north drives a total
distance of 2 blocks, whereas a bird flying “as the crow flies” can fly in
a straight line from start to finish and travel only

√

2 blocks (Euclidean
norm). The Chebyshev norm is simply the absolute value of the vector
component with the largest absolute value. An example of how this norm