The Scalar (Dot) Product — Definition

Scalar Product (Dot Product)

For two vectors $a = (a_{1} \a_{2} \a_{3})$ and $b = (b_{1} \b_{2} \b_{3})$ in three-dimensional space, their scalar product is defined as:
$a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
The result is always a scalar (a real number), not a vector.

The dot between the two vectors ( $\cdot$ ) is the standard notation. Do not confuse this with ordinary multiplication — this is a specific operation between two vectors.

Example

Let $a = 123$ and $b = 456$ . Compute $a \cdot b$ .

$a \cdot b = 1 (4) + 2 (5) + 3 (6) = 4 + 10 + 18 = 32$

Note: $b \cdot a = 4 (1) + 5 (2) + 6 (3) = 32$ — the same result, confirming commutativity.

Note

The scalar product is defined for vectors of any dimension, but in IB DP Mathematics AI HL, you will primarily work with 2D and 3D vectors.

Properties of the Scalar Product

The scalar product obeys a number of algebraic properties that mirror those of ordinary multiplication, with one important addition:

Commutativity: $v \cdot w = w \cdot v$
Distributivity over addition: $u \cdot (v + w) = u \cdot v + u \cdot w$
Scalar multiplication: $(k v) \cdot w = k (v \cdot w)$ for any scalar $k$
Self-dot product: $v \cdot v = ∣ v ∣^{2}$

Property 4 is particularly useful. Since $v \cdot v = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}$ and $∣ v ∣ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}$ , it follows directly that:
$v \cdot v = ∣ v ∣^{2}$

Exam Tip

Property 4 gives you a quick way to find the magnitude of a vector: $∣ v ∣ = v \cdot v$ . This is especially handy when the vector is given as a combination of other vectors.

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AHL 3.13—Scalar and vector products

The Scalar (Dot) Product — Definition

Properties of the Scalar Product

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