Vectors - 3D Math Primer for Graphics and Game Development - page 70

index finger, similar to what we did in Section 1.3.3. Your third finger now
points in the direction of a × b .
Of course, a similar trick works with your right hand for right-handed
coordinate spaces.
One of the most important uses of the cross product is to create a vector
that is perpendicular to a plane (see Section 9.5), triangle (Section 9.6), or
polygon (Section 9.7).

2.13 Linear Algebra Identities

The Greek philosopher Arcesilaus reportedly said, “Where you find the
laws most numerous, there you will find also the greatest injustice.” Well,
nobody said vector algebra was fair. Table 2.2 lists some vector algebra
laws that are occasionally useful but should not be memorized. Several
identities are obvious and are listed for the sake of completeness; all of
them can be derived from the definitions given in earlier sections.

Identity Comments

a + b = b + a Commutative property of vector addition

a − b = a + (−b) Definition of vector subtraction

(a + b) + c = a + (b + c) Associative property of vector addition

s(ta) = (st)a Associative property of scalar multiplication

k(a + b) = ka + kb Scalar multiplication distributes over vector addition

ka = |k|a Multiplying a vector by a scalar scales the magnitude by a factor equal

to the absolute value of the scalar

a ≥ 0 The magnitude of a vector is nonnegative

a 2 + b 2 = a + b 2 The Pythagorean theorem applied to vector addition.

a + b ≥ a + b Triangle rule of vector addition. (No side can be longer than the sum of

the other two sides.)

a b = b a Commutative property of dot product

√

a = a a

Vector magnitude defined using dot product

k(a b) = (ka) b = a (kb) Associative property of scalar multiplication with dot product

a (b + c) = a b + a c Dot product distributes over vector addition and subtraction

a × a = 0 The cross product of any vector with itself is the zero vector. (Because

any vector is parallel with itself.)

a × b = −(b × a) Cross product is anticommutative.

a × b = (−a) × (−b) Negating both operands to the cross product results in the same vector.

k(a × b) = (ka) × b = a × (kb) Associative property of scalar multiplication with cross product.

a × (b + c) = a × b + a × c Cross product distributes over vector addition and subtraction.

Table 2.2
Table of vector algebra identities