Three Dimensional Geometry — Study Notes

1. Introduction

Three-dimensional geometry extends the coordinate geometry of the plane to space by introducing a third coordinate. Lines and planes, which were described by single equations in two dimensions, now become the fundamental objects described by direction cosines, direction ratios, and vector and Cartesian equations. This chapter provides the complete toolkit for working with lines and planes in space.

The chapter begins with direction cosines and direction ratios of a line, the angle between two lines, and then develops the equations of a line in both vector and Cartesian form, including the skew-line concepts of shortest distance. It then introduces the equation of a plane in its various standard forms, the angles between two planes and between a line and a plane, and the distances from a point to a plane.

Much of the material is directly built on the vector algebra of the previous chapter. The dot product computes angles, the cross product provides normals, and the scalar triple product gives distances and volumes. In the board examination, questions from this chapter typically involve converting between vector and Cartesian forms, computing angles, finding distances, and applying the condition for coplanarity of lines. Methodical formula application makes this chapter highly scoreable.

2. Direction Cosines and Direction Ratios

The direction cosines of a line are the cosines of the angles that the line makes with the positive x, y, and z axes. Denoted l, m, n, they satisfy

$$l^2 + m^2 + n^2 = 1$$

If a line has direction ratios a, b, c (any numbers proportional to the direction cosines), then

$$l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}$$

Angle Between Two Lines

If two lines have direction ratios (a1, b1, c1) and (a2, b2, c2), then the cosine of the angle between them is

$$\cos\theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}$$

The lines are perpendicular if $a_1a_2 + b_1b_2 + c_1c_2 = 0$, and parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

3. Equation of a Line in Space

Vector Form

A line through a point with position vector $\vec{a}$ and parallel to a vector $\vec{b}$ is

$$\vec{r} = \vec{a} + \lambda \vec{b}$$

where $\lambda$ is a parameter.

Cartesian Form

If the line passes through $(x_1, y_1, z_1)$ with direction ratios (a, b, c), then

$$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$

4. Distance Between Two Lines

Parallel Lines

If two parallel lines are $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \lambda \vec{b}$, then the shortest distance between them is

$$d = \frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}$$

Skew Lines

If the lines are $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \lambda \vec{b}_2$ (not parallel), the shortest distance is

$$d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$$

The lines are coplanar (intersecting or parallel) if and only if the scalar triple product $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0$.

5. Equation of a Plane

General Form

$$ax + by + cz + d = 0$$

where (a, b, c) is the direction of the normal to the plane.

Vector Form

Through a point with position vector $\vec{a}$ and normal $\vec{n}$:

$$(\vec{r} - \vec{a}) \cdot \vec{n} = 0 \quad \text{or} \quad \vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n} = d$$

Intercept Form

$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$$

where the plane cuts the axes at distances a, b, c. The plane passing through the three points $(\alpha, 0, 0)$, $(0, \beta, 0)$, $(0, 0, \gamma)$.

Planes Through Three Points

The plane through the points with position vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ is

$$(\vec{r} - \vec{a}) \cdot [(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})] = 0$$

Distance from a Point to a Plane

The distance from the point $(x_1, y_1, z_1)$ to the plane $ax + by + cz + d = 0$ is

$$D = \frac{|a x_1 + b y_1 + c z_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$$

6. Angles Involving Planes

Angle Between Two Planes

If the planes are $\vec{r} \cdot \vec{n}_1 = d_1$ and $\vec{r} \cdot \vec{n}_2 = d_2$, then

$$\cos\theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$$

The planes are perpendicular if $\vec{n}_1 \cdot \vec{n}_2 = 0$ and parallel if $\vec{n}_1$ is a scalar multiple of $\vec{n}_2$.

Angle Between a Line and a Plane

If the line has direction vector $\vec{b}$ and the plane has normal $\vec{n}$, then

$$\sin\theta = \frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}||\vec{n}|}$$

Quick Revision Tables

Table 1: Lines in Space

Table 2: Planes in Space

Table 3: Angles

Mind Map

Important Diagrams (SVG)

Diagram 1: Line in Space and Its Direction Cosines

Diagram 2: Distance from a Point to a Plane

Common Mistakes

1. Writing direction cosines as the direction ratios themselves; the direction ratios must be normalised by dividing by sqrt(a^2 + b^2 + c^2).
2. Forgetting that the angle between two lines uses the absolute value of the dot product when acute angles are required.
3. In the Cartesian equation of a line, using the coordinates of a point that does not actually lie on the line.
4. Using the distance formula for skew lines for parallel lines; the formulas differ, and the parallel-line formula divides by |b|, not |b1 x b2|.
5. Forgetting the absolute value in the point-to-plane distance formula; the result must always be positive.
6. In the angle between a line and a plane, using cos instead of sin; the angle between the line and its projection on the plane uses sin theta = |b.n|/(|b||n|).
7. Declaring two planes perpendicular when their normals are parallel, or vice versa; parallel planes have proportional normals.
8. In the intercept form, mixing up the signs of the intercepts when a plane cuts negative axes.
9. Confusing the condition for coplanarity of two lines with the condition for parallelism; coplanar means the shortest distance is zero.
10. Writing the equation of a plane through three points without verifying that the points are non-collinear.

Exam Tips

1. Convert direction ratios into direction cosines by normalising; this is the standard first step in many problems.
2. For the angle between lines, identify the direction vectors from the denominators of the symmetric form of each line.
3. To find the shortest distance between skew lines, apply the formula in one go; if the scalar triple product is zero, the lines are coplanar and the distance is zero.
4. Write the general equation of a plane passing through a given point and a given normal first, then expand to Cartesian form.
5. For the angle between a line and a plane, compute sin theta, not cos theta; verify the angle is acute.
6. Use the intercept form when a plane cuts the axes at known points, and solve for the plane equation by substituting the three points otherwise.
7. When a plane is perpendicular to another, equate the dot product of the normals to zero; when parallel, equate the ratios of the normals.
8. Practise converting vector forms of lines and planes to Cartesian forms and back, since the exam frequently asks for both representations.

Conclusion

Three-dimensional geometry completes the analytic description of space that began with coordinate geometry in two dimensions. Direction cosines and direction ratios describe the orientation of lines, the vector and symmetric Cartesian equations locate them in space, and the shortest-distance formulas settle the relationship between two lines. Planes are described by their normals through several equivalent forms, and angles and distances between lines and planes are computed with a small set of vector formulas that follow directly from the dot and cross products. Because every formula in this chapter is derived from the vector algebra of the preceding chapter, the two chapters must be studied together. With careful formula recall and clean step-by-step presentation, this chapter offers dependable marks and a rigorous foundation for the study of spatial geometry in higher mathematics and engineering.