Introduction to the Vector Product

The vector product (also called the cross product) is a binary operation on two vectors in three-dimensional space. Unlike the dot product, which returns a scalar, the vector product returns a new vector that is perpendicular to both input vectors.

Vector Product

The vector product of two vectors $v$ and $w$ , written $v \times w$ , is a vector with magnitude $∣ v ∣∣ w ∣ sin θ$ and direction perpendicular to both $v$ and $w$ , determined by the right-hand rule.

The vector product is defined only in three dimensions (and seven dimensions, but that's well beyond IB scope). Every application you encounter in IB Maths AA HL will be in $R^{3}$ .

Warning

Do not confuse the dot product and the cross product:

Dot product $a \cdot b$ : returns a scalar, involves $cos θ$
Cross product $a \times b$ : returns a vector, involves $sin θ$

They measure fundamentally different geometric relationships.

The Formula: Geometric Definition

The vector product of $v$ and $w$ is formally defined as:

$v \times w = ∣ v ∣∣ w ∣ sin θ n$

where:

$∣ v ∣$ and $∣ w ∣$ are the magnitudes of the two vectors
$θ$ is the angle between $v$ and $w$ (where $0 \leq θ \leq π$ )
$n$ is the unit normal vector perpendicular to both $v$ and $w$

The Right-Hand Rule determines the direction of $n$ : point your right hand's fingers in the direction of $v$ , curl them toward $w$ through the smaller angle — your thumb points in the direction of $v \times w$ .

Analogy

Think of a screwdriver: if you turn the screw from $v$ toward $w$ , the direction the screw advances is the direction of $v \times w$ . This is the "right-hand screw rule."

Note

Since reversing the order reverses the direction of $n$ , we get $w \times v = - (v \times w)$ . The vector product is anti-commutative, not commutative.

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AHL 3.16—Vector product

Introduction to the Vector Product

The Formula: Geometric Definition

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