The angle between two lines

What you will be able to do

• find the angle between two lines from their direction ratios or direction vectors;
• test whether two lines are parallel or perpendicular;
• find the angle that a line makes with an axis.

Two lines, one angle

Two cables cross over a street. They may never touch, but we can still ask how far one is turned from the other. Imagine sliding both lines, without turning them, until they pass through the origin. The angle between these two new lines is the angle between the original lines. This works even for lines that do not meet.

Two lines make two angles, $\theta$ and $180^\circ - \theta$. By convention we take the acute angle (or $90^\circ$). A-Level and IB use the same convention for the angle between two lines.

The formula

Let the direction ratios be $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$. In vector form these are the vectors $\vec{b}_1$ and $\vec{b}_2$. The dot product of the vectors gives

With direction cosines the lengths are $1$, so $\cos\theta = \lvert l_1l_2 + m_1m_2 + n_1n_2 \rvert$. The modulus bars matter. Without them the formula gives the angle between the two direction vectors, which can be obtuse.

The cross product gives the sine in the same way: $\sin\theta = \dfrac{\lvert \vec{b}_1 \times \vec{b}_2 \rvert}{\lvert \vec{b}_1 \rvert \lvert \vec{b}_2 \rvert}$. Some books use $\tan\theta$ to avoid taking a square root.

Parallel and perpendicular lines

• Perpendicular: $\theta = 90^\circ$, so $a_1a_2 + b_1b_2 + c_1c_2 = 0$.
• Parallel: $\theta = 0^\circ$, so $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$. Equivalently $\vec{b}_1 \times \vec{b}_2 = \vec{0}$.

Notice that "perpendicular" here does not mean that the lines meet. Two skew lines can be perpendicular too. The next course studies such lines.

The angle with an axis

The $x$-axis has direction ratios $(1, 0, 0)$. The line with direction ratios $(a, b, c)$ makes the angle $\alpha$ with it, where $\cos\alpha = \dfrac{\lvert a \rvert}{\sqrt{a^2 + b^2 + c^2}}$. This is the modulus of a direction cosine, which is why the direction cosines are named after the angles with the axes.

A worked example

Find the angle between the lines with direction ratios $(1, 2, 2)$ and $(2, 3, 6)$.

1. The dot product. $1 \times 2 + 2 \times 3 + 2 \times 6 = 20$.
2. The lengths. $\sqrt{1 + 4 + 4} = 3$ and $\sqrt{4 + 9 + 36} = 7$.
3. The cosine. $\cos\theta = \dfrac{20}{21}$.
4. The angle. $\theta = \cos^{-1}\dfrac{20}{21} \approx 17.75^\circ$.

More cases. The lines $(1, 2, 2)$ and $(2, 1, -2)$ have dot product $2 + 2 - 4 = 0$, so they are perpendicular. The lines $(1, 2, 2)$ and $(-2, -4, -4)$ are parallel, because the second is $-2$ times the first. The angle between $(2, 3, 6)$ and the $z$-axis has $\cos\theta = \dfrac{6}{7}$, which is about $31.0^\circ$.

An angle between lines given in vector form. For $\vec{r} = \hat{i} + \lambda(\hat{i} + \hat{j})$ and $\vec{r} = 2\hat{k} + \mu(\hat{j} + \hat{k})$, the directions are $\hat{i} + \hat{j}$ and $\hat{j} + \hat{k}$. The dot product is $1$ and each length is $\sqrt{2}$, so $\cos\theta = \dfrac{1}{2}$ and $\theta = 60^\circ$.

Practise with new numbers

Try it in Python

The program finds the dot product, the cosine and the acute angle for the worked example. It then tests for perpendicular and parallel lines.

Common mistakes

• Dropping the modulus. A negative dot product gives an obtuse angle. The angle between two lines is the acute one.
• Using the points instead of the directions. The angle depends only on the direction ratios.
• Forgetting a root. Both lengths are square roots of a sum of three squares.
• Reading a Cartesian line wrongly. The direction ratios are the denominators, and not the numerators.

Remember these

• $\cos\theta = \dfrac{\lvert a_1a_2 + b_1b_2 + c_1c_2 \rvert}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}$.
• Perpendicular: the dot product of the directions is $0$.
• Parallel: the direction ratios are proportional.
• The angle between a line and the $x$-axis has $\cos\alpha = \lvert l \rvert$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the angle between the lines with direction ratios $(1, 1, 0)$ and $(1, 0, 1)$.

2. For which value of $k$ are the lines with direction ratios $(k, 1, 2)$ and $(3, -4, 5)$ perpendicular?

3. Are the lines with direction ratios $(2, -3, 6)$ and $(-4, 6, -12)$ parallel?

4. A line has direction cosines $\dfrac{1}{3}$, $\dfrac{2}{3}$ and $\dfrac{2}{3}$. Find the acute angle between it and the $y$-axis.

5. Why is the answer to a question on the angle between two lines never greater than $90^\circ$?

References

• OpenStax, Calculus Volume 3, Chapter 2, sections on the dot product and on equations of lines in space.
• NCERT, Mathematics Part II for Class XII, Chapter 11, section on the angle between two lines.