Angles Between Lines

Why Direction Determines the Angle

A line in space has a position and a direction. Its position tells us where it lies, while its direction tells us how it is oriented through space. The angle between two lines depends on their directions, not on where the lines are located.

This means the angle can be found even when the lines do not intersect. Two lines in space may be skew, but their direction vectors still determine the angle between them. Because a line has no preferred forward direction, the usual geometric angle between two lines is the smaller angle, between \(0^\circ\) and \(90^\circ\). When the problem specifies orientations or directed vectors, the directed angle can instead range from \(0^\circ\) to \(180^\circ\).

If orientations are specified, use the directed angle \(\phi\) between the chosen direction vectors:

The directed angle may be obtuse. Reversing one direction vector changes an angle \(\phi\) to \(180^\circ-\phi\), while the smaller angle between the underlying lines remains unchanged.

Finding Direction Vectors

A line in parametric form is written as

where \(\mathbf{r}_0\) is a point on the line, \(t\) is a parameter, and the nonzero vector \(\mathbf{d}\) is a direction vector. In component form, \(\mathbf{r}_0\) determines the fixed coordinates and the coefficients of \(t\) give the direction vector.

For a line given by symmetric equations, such as

the direction vector is \(\mathbf{d}=(a,b,c)\), provided that the displayed equation represents a line and the direction vector is nonzero. A direction vector can be multiplied by any nonzero scalar without changing the line’s direction as a geometric line. In particular, \(\mathbf{d}\) and \(-\mathbf{d}\) represent the same unoriented line direction.

Why the Dot-Product Formula Works

For two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\), let \(\alpha\) be the angle between them, measured from \(0\) to \(\pi\). The dot product satisfies

One way to see why is to compare two expressions for the squared length of \(\mathbf{u}-\mathbf{v}\). Expanding with the dot product gives

The law of cosines, applied to the triangle formed by \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{u}-\mathbf{v}\), gives

Equating these expressions and cancelling the shared squared-length terms yields

Since both vectors are nonzero, their magnitudes are positive, so division by their product is valid. Solving for \(\alpha\) gives the directed-vector formula. For lines, either direction vector may be reversed. Reversing just one changes the sign of the dot product and replaces \(\cos\alpha\) by \(-\cos\alpha\). Taking the absolute value therefore selects the same smaller angle whichever direction representatives are used.

Worked Examples

Orientation and Equivalent Direction Choices

A line’s direction vector is not unique. If \(\mathbf{u}\) is a direction vector, then \(k\mathbf{u}\), for any nonzero real number \(k\), is also a direction vector for the same line. In the angle formula, scaling either vector by a positive number multiplies both the relevant dot product and magnitude by that number, so the ratio is unchanged. Scaling by a negative number reverses its orientation; the absolute value in the line-angle formula again leaves the result unchanged.

The two conventions can be summarized as follows:

If the prompt asks for “the angle between two lines” without specifying orientations, use the smaller angle. If it gives directed vectors, rays, or an explicit orientation and asks for the angle between those directions, retain the sign of the dot product. If the prompt requests an obtuse angle, check that it is asking about oriented directions; the usual smaller angle between unoriented lines cannot be obtuse.

Common Mistake: Keeping the Sign for an Unoriented Line

For the vectors \((1,0,0)\) and \((-1,1,0)\), a common error is to calculate \[ \cos\theta=\frac{-1}{\sqrt{2}} \] and report \(135^\circ\) as the angle between the lines. This calculation gives the angle between the chosen oriented vectors, but a line can equally well use the opposite direction vector.

For the smaller angle between the lines, correct the calculation by using the absolute value:

Another error is to use the position vectors of points on the lines instead of direction vectors. Position vectors locate points; they do not in general describe how the lines run. First identify a nonzero direction vector for each line, then apply the dot-product formula.

Special Cases and a Reliable Procedure

If the direction vectors are parallel or antiparallel, their dot-product ratio has absolute value \(1\), giving a smaller angle of \(0^\circ\). This includes distinct parallel lines and coincident lines. If the direction vectors are perpendicular, their dot product is \(0\), giving an angle of \(90^\circ\). The formula requires nonzero direction vectors; the zero vector cannot represent the direction of a line and cannot be used in the denominator.

This method separates the geometry from the arithmetic. The line positions may affect whether lines meet, but they do not affect the angle determined by their direction vectors. The absolute value is not an optional convenience: it encodes the fact that an unoriented line has no preferred direction.