In a plane, two lines either cross or run parallel. In space there is a third possibility: two lines can miss each other completely without being parallel, like a diagonal across the ceiling and a diagonal down a wall. This chapter uses the vectors of Chapter 10 to describe any line in space with one equation, to measure the angle between two lines, and to find the shortest gap between two lines that never meet.
A line in space makes an angle with each of the three axes. The cosines of those three angles tell you which way it points.
A directed line through the origin makes angles α, β, γ with the positive x, y and z axes. Their cosines l, m, n are the line’s direction cosines.
Any three numbers a, b, c proportional to l, m, n are direction ratios. To get back, divide by the length. m and n work the same way with b and c; take the same sign in all three.
Reversing the line changes the angles like this, so all three signs of l, m, n flip. A line that does not pass through the origin has the same direction cosines as the parallel line through the origin.
How to use it: a line through two points
Angles 90°, 60°, 30° with the axes.
Direction ratios 2, −1, −2.
Line through (−2, 4, −5) and (1, 2, 3).
Direction cosines of the axes.
A(2, 3, −4), B(1, −2, 3), C(3, 8, −11) collinear.
A line in space is fixed by one point and a direction (or by two points). Start at the point, then walk along the direction.
A is a point with position vector a⃗, and the line is parallel to b⃗. Every point P of the line is reached by going to A, then some multiple of b⃗. λ is any real number.
The same line in coordinates. The numerators name the point A(x₁, y₁, z₁); the denominators are the direction ratios of b⃗ = ai + bj + ck. With direction cosines l, m, n in place of a, b, c the same form holds.
This is the parametric form: the vector form, one coordinate at a time. Eliminate λ and you get the Cartesian form.
Here a⃗ and b⃗ are the position vectors of the two points, so b⃗ − a⃗ is the direction from one to the other.
Through (5, 2, −4), parallel to 3i + 2j − 8k.
Two lines need not meet to have an angle between them. Slide parallel copies to the origin and measure the angle between their directions.
b⃗₁ and b⃗₂ are the directions of the two lines. The absolute value | | gives the acute angle θ, since a line has no arrowhead.
The same formula when the lines have direction ratios a₁, b₁, c₁ and a₂, b₂, c₂.
No bottom line is needed: direction cosines already have length 1.
The top of the fraction is 0, so cos θ = 0 and θ = 90°.
The direction ratios are proportional: both lines run in the same direction.
Directions i + 2j + 2k and 3i + 2j + 6k.
DRs 3, 5, 4 and 1, 1, 2.
In a plane, two lines either meet or run parallel. In space there is a third case: lines that do neither.
Lines that are neither parallel nor intersecting are called skew lines. They do not lie in any one plane.
In a room of size 1 × 3 × 2, the ceiling diagonal GE and the line DB, running from a ceiling corner diagonally down a wall, are skew.
The shortest distance, case by case
Two lines that intersect have shortest distance 0.
It is the perpendicular distance from a point of one line to the other line.
The shortest distance joins a point of one line to a point of the other so that the segment is as short as possible. For skew lines this segment is perpendicular to both lines.
Two skew lines have one common perpendicular direction. The shortest distance is the shadow of any joining segment on that direction.
For r⃗ = a⃗₁ + λb⃗₁ and r⃗ = a⃗₂ + μb⃗₂. The common perpendicular has direction b⃗₁ × b⃗₂. Take any S on the first line and T on the second: d is the length of the shadow of ST⃗ = a⃗₂ − a⃗₁ on that direction.
For r⃗ = a⃗₁ + λb⃗ and r⃗ = a⃗₂ + μb⃗ the cross product b⃗ × b⃗ is the zero vector 0⃗. So the skew formula cannot be used. Use the parallelogram instead: area ÷ base.
For skew lines through (x₁, y₁, z₁) and (x₂, y₂, z₂) with direction ratios a₁, b₁, c₁ and a₂, b₂, c₂. The top is this 3 × 3 determinant (take its absolute value); then d = top ÷ bottom.
How to use it
r⃗ = i + j + λ(2i − j + k), r⃗ = 2i + j − k + μ(3i − 5j + 2k).
Parallel lines with b⃗ = 2i + 3j + 6k.
This chapter of the textbook has no multiple-choice questions, so these twelve test the ideas themselves. Tap an option to check it and see the reasoning.
Direction
A line
Angle between two lines
Shortest distance
Direction cosines l, m, n are the cosines of the angles with the axes; direction ratios are any multiple of them. With direction ratios the angle is cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√… √…); parallel lines have proportional ratios. Skew lines are neither parallel nor intersecting.
Each walkthrough draws the lines in 3D with the x, y and z axes: the direction arrow that sets a line's slant, a point running along r⃗ = a⃗ + λb⃗, two directions brought together to measure an angle, and the short green segment that bridges two skew lines. Press Next step (or Play); each step is explained in detail under the controls, with links to the basics it uses.
Cosines from angles, cosines from ratios, the line through two points, the axes, and a collinearity test.
From a point and a direction to the vector and Cartesian equations.
The same formula in vector form and in Cartesian form.
Skew lines via the common perpendicular, and parallel lines via a parallelogram.
A drone climbing on a straight course, cables bracing a pole, two pipes crossing a plant room, a laser aimed at a target, and beams in a ceiling: each is a line in space. Its equation r⃗ = a⃗ + λb⃗ tells you where it goes; dot and cross products tell you the angle between two of them and how close they come.
A drone takes off from A(1, 1, 0) (metres) and flies with constant velocity b⃗ = 2i + 3j + 2k metres per second. After t seconds it is at r⃗ = a⃗ + t b⃗: the parameter λ is simply time. Its direction cosines tell you the climb angle, and the Cartesian form (x − 1)/2 = (y − 1)/3 = z/2 is the flight path.
A 6 m pole stands at the origin. Cable 1 runs from its top to an anchor 4 m along the x-axis; cable 2 to an anchor you can move around the ground. The angle between the cables comes from their direction ratios: cos θ = |a₁a₂ + b₁b₂ + c₁c₂|/(√… √…).
In a plant room, pipe 1 runs along the line r⃗ = 3k + λi (3 m up, along x). Pipe 2 starts at (2, 0, 1) and climbs across the room with direction j + s k. The pipes are skew. The shortest connector joins them along b⃗₁ × b⃗₂, and its length is |(b⃗₁ × b⃗₂) · (a⃗₂ − a⃗₁)|/|b⃗₁ × b⃗₂|.
A laser at the origin shines along the line r⃗ = λ(cos φ i + sin φ j + t k). A target sits at (4, 3, 2). The miss distance is the length of the perpendicular from the target to the beam, found from the foot of the perpendicular (where the dot product with the direction is 0).
A room is L long, 3 m wide and 2 m high. A beam runs diagonally across the ceiling from G to E; a brace runs from D, a ceiling corner, diagonally down the wall to B. They are skew. Change the room's length and watch the gap between them.
Decide first, then tap each card.
All 25 questions of the chapter have an animated walkthrough here (the chapter has no multiple-choice questions). Pick a question from the chips above each player, then press Play or step through with Next. A fourth player, clearly marked Beyond the book, adds four extra problems that use the same ideas: a point on a line, whether two lines meet, the foot of a perpendicular, and a line through two points.
From angles, from equal angles, from ratios, a collinearity test and the sides of a triangle.
Perpendicular and parallel tests, equations of lines, angles between lines, and four shortest distances.
An angle with letters, the x-axis as a line, a perpendicularity condition, a shortest distance and a line perpendicular to two others.
Extra practice, not from the textbook: is a point on a line, do two lines meet, the foot of a perpendicular, and a line through two points.
Lines in space rest on eight older ideas: naming points with three coordinates, the distance formula, direction cosines, lines in a plane, the vector a⃗ + λb⃗, and the dot product, cross product and projection from Chapter 10. Here is each one from its roots. Every example and exercise links here exactly where you need it, and a button takes you straight back to the problem.
Two perpendicular number lines name every point of a plane with two numbers. Space needs a third axis, the z-axis, perpendicular to both. The axes are right-handed: curl your right hand's fingers from x to y and your thumb points along z. The point P(x, y, z) is reached from the origin by walking x along the x-axis, then y parallel to the y-axis, then z parallel to the z-axis. On paper the x-axis is drawn slanting towards you.
To go from P(x₁, y₁, z₁) to Q(x₂, y₂, z₂), the three differences are the edges of a box. Its long diagonal is the distance PQ.
The length of the floor diagonal, by Pythagoras.
The vertical edge is perpendicular to the floor, so Pythagoras again. Those same three differences are also the direction ratios of the line PQ.
Put an arrow of length 1 along the line. Its shadows on the three axes are its direction cosines.
The shadow on the x-axis has length cos α (α is the angle with the x-axis), on the y-axis cos β, and on the z-axis cos γ. These shadows are exactly the arrow’s three coordinates: the arrow is (l, m, n) = (cos α, cos β, cos γ).
The arrow’s length is 1, so Pythagoras gives this.
Any arrow along the line, of any length, has components proportional to l, m, n: those are direction ratios. Dividing direction ratios by their length brings you back to the unit arrow.
A line is one point plus a direction. This way of writing a line works in space too.
How a line in the plane was written in Class 11.
Pick one point (x₁, y₁) on the line and a direction (a, b). Every point of the line has this form for some number t, the parameter.
Solve both for t: the two fractions are equal because both equal t. The slope is b/a. In space we simply add a third coordinate.
a⃗ + λb⃗ means: go to A, then move λ times b⃗. As λ runs over all real numbers, the tip sweeps out a whole line.
The position vector of a point A is the arrow from the origin to A.
Multiplying b⃗ by a number λ stretches it by |λ| and reverses it if λ < 0, but keeps it on the same line.
The tip sweeps out the whole line through A parallel to b⃗: that is why this is the equation of a line. Vectors add component by component, so r⃗ = (x₁ + λa)i + (y₁ + λb)j + (z₁ + λc)k.
From Chapter 10: the dot product is the length of a⃗ times the shadow of b⃗ on it.
The shadow form, and the same number in components.
Solve for the angle. It is positive for acute angles, zero exactly at 90°, and negative for obtuse angles.
For lines we want the acute angle, so we take the absolute value: a line pointing “backwards” is still the same line.
The cross product a⃗ × b⃗ is a vector that stands straight up from both a⃗ and b⃗.
A vector perpendicular to both a⃗ and b⃗ (right-hand rule).
The area of the parallelogram on a⃗ and b⃗. The cross product is 0⃗ exactly when a⃗ and b⃗ are parallel.
b⃗₁ × b⃗₂ gives the direction of the shortest segment between two lines, and this (area ÷ base) is the gap between parallel lines.
A segment can change length while its shadow on one direction stays the same. That is the key idea behind the shortest-distance formula.
The projection (shadow) of a vector w⃗ on a direction n̂ of length 1 has this length.
Drawn in a plane: two parallel lines 2 units apart. Join any point S on one to any point T on the other. ST⃗ can be long or short, but its shadow on the direction straight across (n̂) is always the gap, 2.
For skew lines the “straight across” direction is b⃗₁ × b⃗₂, so this holds whichever S and T you pick.
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