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CBSE · Class 12 · Mathematics

Equation of a Line in Space (Ch 11: 11.3)

2 concepts · 16 practice questions with answers · Free to read

Vector and Cartesian forms

From daily life

To travel in a straight line, you need a starting point and a direction. All points of the road are found by walking some number λ of steps in that direction from the start.

Simple definition

A line through the point with position vector a and parallel to the vector b has the vector equation r = a + λb, where λ is any real number. If the point is (x₁, y₁, z₁) and b = ai + bj + ck, the Cartesian form is (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.

Formula: r = a + λb, (x − x₁)/a = (y − y₁)/b = (z − z₁)/c

Rule 1: Point and direction — Write the point and the direction first.

  • Easy: The line through (1, 2, 3) parallel to 2i + j + k is r = (i + 2j + 3k) + λ(2i + j + k), and in Cartesian form (x − 1)/2 = (y − 2)/1 = (z − 3)/1.
  • Medium: From (x − 2)/3 = (y + 1)/2 = (z − 5)/6, the line passes through (2, −1, 5) with direction ratios 3, 2, 6.
  • Exam-type: A point lies on the line if one value of λ fits all three coordinates. For (5, 1, 11) on the line above, all three ratios equal 1, so it lies on the line.

Rule 2: Switching forms — Read off the point and the ratios.

  • Easy: The line through the origin parallel to i + j + k is r = λ(i + j + k).
  • Medium: The line through (1, −1, 2) parallel to the y-axis has x = 1 and z = 2.
  • Exam-type: The line through (2, 1, −1) parallel to r = i + 2j + 3k + λ(4i − j + 2k) is (x − 2)/4 = (y − 1)/(−1) = (z + 1)/2.
Remember

A zero in the denominator means that coordinate stays constant. Write it as x = x₁ separately.

Common mistake

Writing (y + 1) with the wrong sign when the point has a negative coordinate.

Memory clue

Point plus a multiple of direction.

Practice questions

Easy
  1. 1Write the Cartesian form of the line through (1, 2, 3) with direction ratios 2, 1, 1.
  2. 2Write the vector equation of the line through the origin parallel to i + j + k.
  3. 3From (x − 2)/3 = (y + 1)/2 = (z − 5)/6, find a point and the direction ratios.
Medium
  1. 4Does (5, 1, 11) lie on (x − 2)/3 = (y + 1)/2 = (z − 5)/6?
  2. 5Does (1, 1, 1) lie on the same line?
  3. 6Write the equation of the line through (1, −1, 2) parallel to the y-axis.
Challenging
  1. 7Find the line through (2, 1, −1) parallel to r = i + 2j + 3k + λ(4i − j + 2k).
  2. 8Find the direction cosines of the line (x − 2)/3 = (y + 1)/2 = (z − 5)/6.
Show answers (try them yourself first)
  1. 1(x − 1)/2 = (y − 2)/1 = (z − 3)/1
  2. 2r = λ(i + j + k)
  3. 3Point (2, −1, 5), direction ratios 3, 2, 6
  4. 4Yes, since (5 − 2)/3 = (1 + 1)/2 = (11 − 5)/6 = 1
  5. 5No, since (1 − 2)/3 = −1/3 is not equal to (1 + 1)/2 = 1
  6. 6x = 1 and z = 2 with y free, that is (x − 1)/0 = (y + 1)/1 = (z − 2)/0
  7. 7(x − 2)/4 = (y − 1)/(−1) = (z + 1)/2
  8. 83/7, 2/7, 6/7

Line through two points

From daily life

Two pegs fixed in the ground fix a straight string. In space too, two points are enough to fix the line completely.

Simple definition

The line through A(x₁, y₁, z₁) and B(x₂, y₂, z₂) has direction ratios x₂ − x₁, y₂ − y₁, z₂ − z₁ and passes through A. Using these in the Cartesian form gives its equation. In vector form, r = a + λ(b − a).

Formula: (x − x₁)/(x₂ − x₁) = (y − y₁)/(y₂ − y₁) = (z − z₁)/(z₂ − z₁)

Rule 1: Subtract to get the direction — Then use either point.

  • Easy: Through (1, 2, 3) and (3, 5, 9): (x − 1)/2 = (y − 2)/3 = (z − 3)/6.
  • Medium: Through (2, 3, 1) and (4, 3, 5): direction ratios 2, 0, 4, so y = 3 and (x − 2)/2 = (z − 1)/4.
  • Exam-type: The line through (1, 2, 3) and (3, 5, 9) meets the xy-plane (z = 0) at λ = −1/2 on the line (x, y, z) = (1 + 2λ, 2 + 3λ, 3 + 6λ), which is (0, ½, 0).

Rule 2: Test and convert — Find a parameter value for each point.

  • Easy: (5, 8, 15) lies on the line above, since all three ratios are 2.
  • Medium: The origin lies on the line through (1, 2, 3) and (2, 4, 6), at λ = −1 from (1, 2, 3).
  • Exam-type: The line r = (2i − j + k) + λ(i + 2j − k) is (x − 2)/1 = (y + 1)/2 = (z − 1)/(−1).
Remember

A plane meets a line when you set the matching coordinate to zero and find λ.

Common mistake

Mixing the order of the two points for different coordinates.

Memory clue

Subtract for direction, any point for the position.

Practice questions

Easy
  1. 1Find the line through (1, 2, 3) and (3, 5, 9) in Cartesian form.
  2. 2Find the line through (2, 3, 1) and (4, 3, 5).
  3. 3Find where the line through (1, 2, 3) and (3, 5, 9) meets the plane z = 0.
Medium
  1. 4Write the vector equation of the line through i + j and 3i + 2j + k.
  2. 5Does C(5, 8, 15) lie on the line through (1, 2, 3) and (3, 5, 9)?
  3. 6Does the origin lie on the line through (1, 2, 3) and (2, 4, 6)?
Challenging
  1. 7Find the point on (x − 1)/2 = (y − 2)/3 = (z − 3)/6 at λ = 2.
  2. 8Write r = (2i − j + k) + λ(i + 2j − k) in Cartesian form.
Show answers (try them yourself first)
  1. 1(x − 1)/2 = (y − 2)/3 = (z − 3)/6
  2. 2y = 3 and (x − 2)/2 = (z − 1)/4
  3. 3(z − 3)/6 = −1/2, so the point is (0, ½, 0)
  4. 4r = (i + j) + λ(2i + j + k)
  5. 5Yes, since (5 − 1)/2 = (8 − 2)/3 = (15 − 3)/6 = 2
  6. 6Yes, since (0 − 1)/1 = (0 − 2)/2 = (0 − 3)/3 = −1
  7. 7(1 + 4, 2 + 6, 3 + 12) = (5, 8, 15)
  8. 8(x − 2)/1 = (y + 1)/2 = (z − 1)/(−1)