How Do You Determine If Two Vectors Are Perpendicular or Parallel Using the Scalar Product?

Two non-zero vectors are perpendicular when their scalar product is zero. They are parallel when the absolute value of their scalar product equals the product of their magnitudes.

The scalar product connects algebra to the angle between vectors:

$$\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta,$$

where $\theta$ is the angle between them. In component form, calculate

$$\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+a_3b_3.$$

This allows the relationship to be tested without first finding the angle.

Use this three-step method:

1. Confirm that neither vector is the zero vector, because its direction and angle are undefined.
2. Compute the scalar product from corresponding components.
3. Compare the result with zero for perpendicularity, or compare its absolute value with the product of the magnitudes for parallelism.

For parallelism, a second valid method is to show $\mathbf{b}=k\mathbf{a}$ for some non-zero scalar $k$. The sign of a non-zero scalar product only indicates whether the angle is acute or obtuse; it does not by itself establish parallelism.

For perpendicular vectors, a zero scalar product forces $\cos\theta=0$, so the angle is $90^\circ$. For parallel vectors, $|\cos\theta|=1$, giving the combined test

$$|\mathbf{a}\cdot\mathbf{b}|=|\mathbf{a}||\mathbf{b}|.$$

For example, let $\mathbf{a}=(1,2,-1)$ and $\mathbf{b}=(2,-1,0)$. Then

$$\mathbf{a}\cdot\mathbf{b}=1(2)+2(-1)+(-1)(0)=0.$$

Therefore, $\mathbf{a}$ and $\mathbf{b}$ are perpendicular. The conclusion must follow the calculation; simply obtaining zero without interpreting it is incomplete.

Now let $\mathbf{c}=(-3,-6,3)$. Since

$$\mathbf{a}\cdot\mathbf{c}=-18$$

and

$$|\mathbf{a}||\mathbf{c}|=\sqrt{6}\times3\sqrt{6}=18,$$

we have $|\mathbf{a}\cdot\mathbf{c}|=|\mathbf{a}||\mathbf{c}|$. The vectors are parallel, and the negative scalar product shows that they point in opposite directions. Equivalently, $\mathbf{c}=-3\mathbf{a}$, which directly confirms parallelism.

A common misconception is that any negative scalar product proves parallelism. It only shows that the angle is obtuse. Parallelism requires equality between the absolute scalar product and the magnitude product.

Exam technique: This is Mathematics AA HL, AHL 3.13: Scalar product. On Paper 1, show the component calculation, state the relevant condition, and give a written conclusion. These tests require non-zero vectors because the angle involving the zero vector is undefined.