- Chapter 11
- Vectors and Three-Dimensional Geometry
- 20 questions
Three Dimensional Geometry
Direction cosines and ratios of a line, equations of a line in vector and Cartesian form, the angle between two lines, and skew lines with the shortest distance.
In two dimensions two lines either meet or are parallel. In three dimensions there is a third possibility: they can miss each other entirely without being parallel. Such lines are called skew, and the shortest distance between them is one of the most frequently examined results in the course.
The chapter rests on the previous one. A line in space is described by a point on it together with a direction vector, and every formula here follows from the dot and cross products.
What you should be able to do
- Find the direction cosines and direction ratios of the line joining two points.
- Write the equation of a line in both vector and Cartesian form, and convert between them.
- Find the angle between two lines.
- Identify skew lines and find the shortest distance between them.
Topics
4 topics
- Direction Cosines and Ratios of a Line
The direction of a line joining two points, expressed as cosines and as ratios.
- Equation of a Line
Vector and Cartesian equations of a line, through a point in a given direction and through two points.
- Angle Between Two Lines
Finding the angle from direction ratios, and the conditions for parallel and perpendicular lines.
- Skew Lines and Shortest Distance
Identifying skew lines, and the shortest distance between two skew or two parallel lines.
Formulas
10 items · All formulas
Direction cosines of a line
l^{2} + m^{2} + n^{2} = 1Direction cosines are the components of a unit vector along the line.
Direction ratios from two points
\left(x_2 - x_1,\; y_2 - y_1,\; z_2 - z_1\right)Direction ratios of the line joining (x_1,y_1,z_1) and (x_2,y_2,z_2).
Direction cosines from direction ratios
l = \dfrac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, \quad m = \dfrac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, \quad n = \dfrac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}Vector equation of a line
\vec{r} = \vec{a} + \lambda\vec{b}Through the point with position vector \vec{a}, in the direction \vec{b}.
Cartesian equation of a line
\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}Through (x_1,y_1,z_1) with direction ratios a, b, c.
Line through two points
\vec{r} = \vec{a} + \lambda\left(\vec{b} - \vec{a}\right)Angle between two lines
\cos\theta = \left|\dfrac{\vec{b_1}\cdot\vec{b_2}}{\left|\vec{b_1}\right|\left|\vec{b_2}\right|}\right|The modulus gives the acute angle, which is the convention for the angle between lines.
Perpendicular and parallel conditions
\text{Perpendicular: } a_1a_2 + b_1b_2 + c_1c_2 = 0; \qquad \text{Parallel: } \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}Shortest distance between two skew lines
d = \left|\dfrac{\left(\vec{a_2}-\vec{a_1}\right)\cdot\left(\vec{b_1}\times\vec{b_2}\right)}{\left|\vec{b_1}\times\vec{b_2}\right|}\right|- Valid when
- Requires \vec{b_1} \not\parallel \vec{b_2}.
Distance between two parallel lines
d = \dfrac{\left|\vec{b}\times\left(\vec{a_2}-\vec{a_1}\right)\right|}{\left|\vec{b}\right|}- Valid when
- Both lines share the direction \vec{b}.
Key points
6 items · All key points
Two lines in space may intersect, be parallel, or be skew — skew is the case with no plane containing both.
If the shortest distance between two lines comes out as zero, the lines intersect.
Check whether the direction vectors are parallel before choosing a distance formula.
The skew formula divides by \left|\vec{b_1}\times\vec{b_2}\right|, which is zero for parallel lines.
Use the modulus in the angle formula — the angle between two lines is taken to be acute.
In the Cartesian form, the denominators are the direction ratios; the numerators must be x - x_1, not x + x_1.
Direction ratios are not unique — any non-zero multiple describes the same direction.
Common mistakes
5 items · All common mistakes
- Mistake
Applying the skew-lines formula to two parallel lines.
InsteadFor parallel lines \vec{b_1}\times\vec{b_2} = \vec{0}, so use d = \frac{\left|\vec{b}\times(\vec{a_2}-\vec{a_1})\right|}{|\vec{b}|} instead.
WhyThe formula is applied before the relationship between the lines is established.
- Mistake
Reading the point on the line with the wrong sign from the Cartesian form.
InsteadIn \frac{x+2}{3} = \frac{y-1}{4} = \frac{z}{5}, the point is (-2, 1, 0) — set each numerator to zero.
WhyThe standard form has minus signs, so a written plus sign is easily copied as a positive coordinate.
- Mistake
Omitting the modulus in the angle formula and reporting an obtuse angle.
InsteadThe angle between two lines is the acute one; take the absolute value of the cosine.
- Mistake
Using position vectors where direction vectors are required in the distance formula.
Instead\vec{a_1}, \vec{a_2} are points on the lines; \vec{b_1}, \vec{b_2} are the directions. Label them before substituting.
- Mistake
Treating direction ratios as though they were direction cosines.
InsteadDivide the ratios by \sqrt{a^2+b^2+c^2} to obtain direction cosines, then check l^2+m^2+n^2=1.