How to find the acute angle between two lines

Two intersecting lines can make a wide angle and a narrow one. We’ll use the dot product to find the smaller angle, even when the vectors we choose point toward the wider opening.

In the video, the vectors taken from the coefficients are perpendicular to the lines. Using two such normal vectors gives the same smaller angle as using two direction vectors along the lines.

Pick two direction vectors or two normal vectors

For a line in the form $a x + b y = c$ , the vector $⟨ a, b ⟩$ is a normal vector: it’s perpendicular to the line. A direction vector along the line is $⟨ b, - a ⟩$ . Use two normals or two directions. Rotating both normals by the same quarter turn gives two directions and preserves their angle.

Use the dot product to get the smaller angle

For two nonzero vectors $u$ and $v$ , the angle $θ$ between the arrows has cosine $(u \cdot v) / (| u | | v |)$ . A line doesn’t have a preferred direction. Flipping a vector changes the sign of the dot product, so take the absolute value to get the smaller angle $φ$ between the lines.

\cos φ = \frac{| u \cdot v |}{| u | | v |}

Take the inverse cosine of that number. The smaller angle is between $0^{°}$ and $90^{°}$ , inclusive. Parallel lines give $0^{°}$ and perpendicular lines give $90^{°}$ . Neither of those is strictly acute.

Example: Find the angle from two equations

Use the lines $x + 3 y = 2$ and $3 x - 6 y = 5$ . Their normal vectors are $⟨ 1, 3 ⟩$ and $⟨ 3, - 6 ⟩$ . The dot product is $3 - 18 = - 15$ , and the lengths are $\sqrt{10}$ and $\sqrt{45}$ .

\begin{array}{l} \cos φ = \frac{15}{\sqrt{10} \sqrt{45}} \\ = \frac{1}{\sqrt{2}} \\ φ = 45^{°} \end{array}

Use the smaller opening between the lines. The opposite opening is also 45°.

Without the absolute value, the cosine is negative and the angle between the chosen normal arrows is $135^{°}$ . Its supplement, $180^{°} - 135^{°} = 45^{°}$ , gives the same smaller line angle.

If the question gives you arrows and asks for their actual angle, see how to find the angle between two vectors.

How to find the acute angle between two lines

Pick two direction vectors or two normal vectors

Use the dot product to get the smaller angle

Example: Find the angle from two equations

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