Lines in Space: Quick Revision

A brief note on Lines in Space. Read it before a test. Try each question at the end before you open its answer.

Direction

• Direction cosines: $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$, the cosines of the angles with the three axes. They satisfy $l^2 + m^2 + n^2 = 1$.
• Two sets: a line has $(l, m, n)$ and $(-l, -m, -n)$, one for each direction.
• Direction ratios: any numbers $a$, $b$, $c$ in proportion to $l$, $m$, $n$. Then $l = \dfrac{a}{\sqrt{a^2 + b^2 + c^2}}$, and the same for $m$ and $n$.
• Through two points: the direction ratios are the differences of the coordinates.

Equations of a line

• Vector form: $\vec{r} = \vec{a} + \lambda\vec{b}$, through the point $\vec{a}$ and parallel to $\vec{b}$.
• Cartesian form: $\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}$. A zero denominator means that coordinate stays fixed.
• Through two points: use the differences of the coordinates as direction ratios.

The angle between two lines

• Formula: $\cos\theta = \dfrac{\lvert a_1a_2 + b_1b_2 + c_1c_2 \rvert}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}$. The angle is acute.
• Perpendicular: the dot product of the directions is $0$.
• Parallel: the direction ratios are proportional.

A worked example

Find the acute angle between the lines $\dfrac{x - 1}{3} = \dfrac{y}{4} = \dfrac{z + 4}{12}$ and $\dfrac{x}{1} = \dfrac{y - 2}{2} = \dfrac{z}{2}$.

1. The directions. $(3, 4, 12)$ and $(1, 2, 2)$.
2. The dot product. $3 + 8 + 24 = 35$.
3. The lengths. $13$ and $3$.
4. The cosine. $\cos\theta = \dfrac{35}{39}$, and $\theta \approx 26.18^\circ$.

Practice with new numbers

A program to try

The program finds the direction cosines of the first line and the angle between the two lines of the worked example.

Mistakes to avoid

• Stopping at the ratios. Divide by the length to get direction cosines.
• Dropping the modulus. The angle between two lines is the acute one.
• Reading the point with the wrong sign. In $\dfrac{y + 1}{3}$ the coordinate is $y_1 = -1$.

Check yourself

1. Can $\dfrac{1}{3}$, $\dfrac{2}{3}$ and $\dfrac{2}{3}$ be direction cosines of a line?

2. Find the direction cosines of the line through $(0, 0, 0)$ and $(4, 4, 7)$.

3. Write the line through $(2, 0, -1)$ parallel to $\hat{i} + 3\hat{j} - 2\hat{k}$ in Cartesian form.

4. Are the lines with direction ratios $(2, 1, -2)$ and $(1, 2, 2)$ perpendicular?

5. A student finds a negative cosine for the angle between two lines. What should she do?

Sources

• OpenStax, Calculus Volume 3, Chapter 2, section on equations of lines and planes in space.
• NCERT, Mathematics Part II for Class XII, Chapter 11, sections on direction cosines, the equation of a line and the angle between two lines.