Read the scalar product and its laws

Read the scalar product and its laws

A dot product returns a number. Use it to connect two vector lengths with their angle, or to test perpendicularity. Check that both vectors are nonzero before interpreting a zero product as a right angle.

The scalar product, or dot product, of nonzero vectors is

$$\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta,\qquad0\le\theta\le\pi.$$

It produces a scalar, not a vector.

To read $\theta$, translate the free vectors without rotating them so both point away from a common vertex or both point towards it.

Reversing both preserves the dot product; reversing just one changes its sign and usually gives the supplementary angle.

This is the key arrow-choice guidance repeated beside the source definition and Example 13. The source QR for exploring this geometry decodes to the CJC mathematics learning resource.

Use the product laws with their conditions

For nonzero $\mathbf a,\mathbf b$, the six source properties are:

1. Self-product. $\mathbf a\cdot\mathbf a=|\mathbf a|^2$, since $|\mathbf a||\mathbf a|\cos0=|\mathbf a|^2$. In particular, $(2\mathbf a+\mathbf b)\cdot(2\mathbf a+\mathbf b)=|2\mathbf a+\mathbf b|^2$, the additional example supplied in the Teacher note. The self-product identity also holds when its vector is zero.
2. Parallel directions. $|\mathbf a\cdot\mathbf b|=|\mathbf a||\mathbf b|$ if and only if the directions are parallel. Absolute-value bars are essential: $|\cos\theta|=1$ gives $\theta=0$ or $\pi$. Conversely those angles have cosines $1,-1$, giving dot products $|\mathbf a||\mathbf b|$ and $-|\mathbf a||\mathbf b|$, respectively, with equal absolute value.
3. Perpendicular directions. $\mathbf a\cdot\mathbf b=0$ if and only if $\theta=\pi/2$: divide $|\mathbf a||\mathbf b|\cos\theta=0$ by the nonzero lengths, then use $\cos\theta=0$. Conversely $\cos(\pi/2)=0$ makes the product zero. A zero vector also has zero dot product with every vector, but it has no defined angle; do not infer a geometric angle for it.
4. Commutativity. $\mathbf a\cdot\mathbf b=\mathbf b\cdot\mathbf a$: both are the same product $|\mathbf a||\mathbf b|\cos\theta$.
5. Compatibility with scalar multiplication. The source proves the following chain for $k>0$, when scaling preserves the direction:

$$\begin{aligned} \mathbf a\cdot(k\mathbf b)&=|\mathbf a||k\mathbf b|\cos\theta\\ &=k|\mathbf a||\mathbf b|\cos\theta\\ &=k(\mathbf a\cdot\mathbf b)\\ &=|k\mathbf a||\mathbf b|\cos\theta=(k\mathbf a)\cdot\mathbf b. \end{aligned}$$

The algebraic identity holds for every real $k$: for negative $k$, both direction reversal and magnitude scaling must be included; for zero $k$, every term is zero. The source calls this an associative law, but it is not a rule for dotting three vectors. For example, the Teacher’s unit-direction calculation gives

$$\mathbf a\cdot\widehat{\mathbf b}=\mathbf a\cdot\frac{\mathbf b}{|\mathbf b|}=\frac1{|\mathbf b|}(\mathbf a\cdot\mathbf b),\qquad\mathbf b\ne\mathbf0.$$

6. Distributivity. $\mathbf a\cdot(\mathbf b+\mathbf c)=\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c$. The source defers its geometric proof to the length-of-projection section (7.4.3). Use this law with commutativity to expand either factor in a product of sums.