Revision notes, worked problems, a question bank, assertion and reason questions and answers to the NCERT exercises. Tap a group to open it; opening one closes the others.
If a directed line through the origin makes angles α, β, γ (the direction angles) with the positive x, y and z axes, then l = cos α, m = cos β, n = cos γ are its direction cosines.
Any three numbers a, b, c proportional to l, m, n are direction ratios: a = λl, b = λm, c = λn for some nonzero λ.
For P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) the direction ratios are x₂ − x₁, y₂ − y₁, z₂ − z₁ and the direction cosines are these divided by PQ = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²).
The line through the point with position vector a and parallel to b has
The line through A(a) and B(b) has direction b − a:
Through (5, 2, −4), parallel to 3î + 2ĵ − 8k̂:
A zero direction ratio is allowed. For example the x-axis is x/1 = y/0 = z/0, which means y = 0 and z = 0.
If two lines have direction ratios a₁, b₁, c₁ and a₂, b₂, c₂, the (acute) angle θ between them satisfies
Lines with ratios (1, 2, 2) and (3, 2, 6): dot product = 3 + 4 + 12 = 19, lengths 3 and 7, so cos θ = 19/21.
For r = a₁ + λb₁ and r = a₂ + μb₂:
For r = a₁ + λb and r = a₂ + μb:
| Want | Use |
|---|---|
| Direction cosines from ratios a, b, c | a, b, c divided by √(a² + b² + c²) |
| Ratios from two points | Subtract the coordinates |
| Line through point and parallel to b | r = a + λb |
| Perpendicular lines | a₁a₂ + b₁b₂ + c₁c₂ = 0 |
| Parallel lines | proportional ratios |
| Skew distance | |(b₁ × b₂) · (a₂ − a₁)|/|b₁ × b₂| |
| Parallel distance | |b × (a₂ − a₁)|/|b| |
Try each problem first, then open the solution to check your working.
Find the direction cosines of a line with direction ratios −4, 8, 8.
√(16 + 64 + 64) = 12.
Divide each ratio by 12.
Answer: −1/3, 2/3, 2/3 (or the negatives of all three)
Find the direction cosines of the line through A(1, −2, 2) and B(3, 2, −2).
Ratios AB = (2, 4, −4) and |AB| = 6.
Answer: 1/3, 2/3, −2/3 (or the negatives)
A line makes angles 60° with the x-axis and 60° with the y-axis. Find the angle it can make with the z-axis.
l = cos 60° = 1/2 and m = cos 60° = 1/2, so n² = 1 − 1/4 − 1/4 = 1/2.
n = ±1/√2, so γ = 45° or 135°.
Answer: 45° or 135°
Show that A(1, 0, −2), B(3, 3, 2) and C(−1, −3, −6) are collinear.
AB = (2, 3, 4) and AC = (−2, −3, −4) = −AB.
AC is a multiple of AB with A common, so the three points lie on one line (C is on the other side of A).
Answer: collinear
Show that the triangle with vertices O(0, 0, 0), B(1, 2, 2) and C(2, 1, −2) has a right angle at O.
OB = (1, 2, 2) and OC = (2, 1, −2).
1(2) + 2(1) + 2(−2) = 0, so OB ⟂ OC.
Answer: angle at O is 90°
Find the vector and Cartesian equations of the line through (3, −1, 2) parallel to 2î − 3ĵ + 5k̂.
Use r = a + λb and (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
Answer: r = 3î − ĵ + 2k̂ + λ(2î − 3ĵ + 5k̂); (x − 3)/2 = (y + 1)/(−3) = (z − 2)/5
Find the Cartesian equation of the line through (2, 3, −1) and (4, −1, 5).
Direction ratios (2, −4, 6), proportional to (1, −2, 3).
Answer: (x − 2)/2 = (y − 3)/(−4) = (z + 1)/6
Write the vector form of (x + 1)/2 = (y − 3)/(−1) = z/4.
The line passes through (−1, 3, 0) and has ratios 2, −1, 4.
Answer: r = −î + 3ĵ + λ(2î − ĵ + 4k̂)
Find the Cartesian equation of the line through (1, 1, 1) parallel to (x − 2)/3 = (y + 1)/(−2) = (z − 4)/6.
Parallel lines share direction ratios 3, −2, 6.
Answer: (x − 1)/3 = (y − 1)/(−2) = (z − 1)/6
Show that (5, −2, 13) lies on the line r = î + 2ĵ + 3k̂ + λ(2î − 2ĵ + 5k̂), and find λ.
Solve 1 + 2λ = 5, 2 − 2λ = −2, 3 + 5λ = 13.
Each gives λ = 2, so the point lies on the line.
Answer: λ = 2
Find the angle between the lines with direction ratios (1, −1, 2) and (2, 1, −1).
Dot product = 1(2) + (−1)(1) + 2(−1) = −1. Lengths √6 and √6.
cos θ = |−1|/6 = 1/6.
Answer: θ = cos⁻¹(1/6)
Show that the lines with direction ratios (2, −1, 3) and (1, −1, −1) are perpendicular.
2(1) + (−1)(−1) + 3(−1) = 2 + 1 − 3 = 0.
Answer: perpendicular
For what value of λ are the lines with direction ratios λ, 2, 3 and 2, 1, λ perpendicular?
2λ + 2 + 3λ = 0, so 5λ = −2.
Answer: λ = −2/5
A directed line has direction ratios (2, −1, 2). Find the angles it makes with the axes.
|d| = 3, so l = 2/3, m = −1/3, n = 2/3.
Answer: α = cos⁻¹(2/3), β = cos⁻¹(−1/3), γ = cos⁻¹(2/3)
Find the angle between lines with direction ratios 1, 1, 1 and 1, 1, 0.
Dot product = 2. Lengths √3 and √2.
cos θ = 2/√6 = √6/3.
Answer: θ = cos⁻¹(√6/3)
Find the shortest distance between r = λ(î + 2ĵ + 3k̂) and r = 2î + k̂ + μ(2î + ĵ + k̂).
b₁ × b₂ = −î + 5ĵ − 3k̂, |b₁ × b₂| = √35.
a₂ − a₁ = 2î + k̂, so (b₁ × b₂) · (a₂ − a₁) = −5.
d = |−5|/√35 = √35/7.
Answer: √35/7
Find the shortest distance between (x − 1)/2 = (y − 2)/3 = (z − 3)/4 and (x − 2)/3 = (y − 4)/4 = (z − 5)/5.
b₁ = (2, 3, 4), b₂ = (3, 4, 5): b₁ × b₂ = −î + 2ĵ − k̂, |b₁ × b₂| = √6.
a₂ − a₁ = (1, 2, 2), so (b₁ × b₂) · (a₂ − a₁) = −1 + 4 − 2 = 1.
d = 1/√6.
Answer: √6/6
Find the distance between the parallel lines r = λ(î + 2ĵ + 2k̂) and r = 3î + μ(î + 2ĵ + 2k̂).
b = (1, 2, 2), a₂ − a₁ = (3, 0, 0). b × (a₂ − a₁) = 6ĵ − 6k̂, magnitude 6√2.
|b| = 3, so d = 6√2/3.
Answer: 2√2
Show that (x − 1)/2 = (y − 2)/3 = (z − 3)/4 and (x − 4)/5 = (y − 1)/2 = z/1 intersect, and find the point of intersection.
b₁ × b₂ = (−5, 18, −11) ≠ 0 and (b₁ × b₂) · (a₂ − a₁) = (−5)(3) + 18(−1) + (−11)(−3) = 0, so the shortest distance is 0.
Put the points equal: 1 + 2s = 4 + 5t, 2 + 3s = 1 + 2t, 3 + 4s = t. The last two give s = −1, t = −1, and the first holds.
The point is (−1, −1, −1).
Answer: (−1, −1, −1)
Find k so that (x − 1)/2 = (y + 1)/3 = (z − 1)/4 and (x − 3)/1 = (y − k)/2 = z/1 intersect.
First line: (1 + 2s, −1 + 3s, 1 + 4s). Second line: (3 + t, k + 2t, t).
From x and z: 1 + 2s = 3 + t and 1 + 4s = t, so s = −3/2 and t = −5.
From y: −1 + 3s = k + 2t gives −11/2 = k − 10, so k = 9/2.
Answer: k = 9/2
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Answers to every exercise question of the NCERT chapter. Try each one first, then tap Show answer.
l = cos 90° = 0, m = cos 135° = −1/√2, n = cos 45° = 1/√2.
Check: 0 + 1/2 + 1/2 = 1.
Answer: l = 0, m = −1/√2, n = 1/√2
Equal angles give l = m = n. Since l² + m² + n² = 1, 3l² = 1, so l = ±1/√3.
Answer: 1/√3, 1/√3, 1/√3 (or all three negative for the opposite direction)
√(18² + 12² + 4²) = √(324 + 144 + 16) = √484 = 22.
Divide each ratio by 22.
Answer: −9/11, 6/11, −2/11 (or the negatives of all three)
AB has direction ratios −3, −5, −3. BC has direction ratios 6, 10, 6.
BC = −2 × AB, so the ratios are proportional and AB ∥ BC. They share the point B, so A, B, C lie on one line.
Answer: Shown
Let A(3, 5, −4), B(−1, 1, 2), C(−5, −5, −2). AB = (−4, −4, 6), BC = (−4, −6, −4), CA = (8, 10, −2).
|AB| = |BC| = √68 = 2√17 and |CA| = √168 = 2√42.
AB: −2/√17, −2/√17, 3/√17. BC: −2/√17, −3/√17, −2/√17. CA: 4/√42, 5/√42, −1/√42.
Answer: AB: −2/√17, −2/√17, 3/√17; BC: −2/√17, −3/√17, −2/√17; CA: 4/√42, 5/√42, −1/√42 (signs reverse for the opposite direction)
Two lines are perpendicular when l₁l₂ + m₁m₂ + n₁n₂ = 0.
Lines 1 and 2: (48 − 36 − 12)/169 = 0.
Lines 2 and 3: (12 − 48 + 36)/169 = 0.
Lines 3 and 1: (36 + 12 − 48)/169 = 0.
Answer: All three pairs give 0, so the lines are mutually perpendicular
Direction ratios of the first line: (2, 5, −4). Of the second: (3, 2, 4).
Sum of products: 2(3) + 5(2) + (−4)(4) = 6 + 10 − 16 = 0.
Answer: Shown
Direction ratios of the first line: (−2, −4, −4). Of the second: (2, 4, 4).
−2/2 = −4/4 = −4/4 = −1, so the ratios are proportional and the lines are parallel.
Answer: Shown
Point a = î + 2ĵ + 3k̂ and direction b = 3î + 2ĵ − 2k̂. Use r = a + λb.
Cartesian form: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
Answer: r = î + 2ĵ + 3k̂ + λ(3î + 2ĵ − 2k̂); Cartesian: (x − 1)/3 = (y − 2)/2 = (z − 3)/(−2)
a = 2î − ĵ + 4k̂ and b = î + 2ĵ − k̂, so r = a + λb. The point is (2, −1, 4) with direction ratios 1, 2, −1.
Answer: r = 2î − ĵ + 4k̂ + λ(î + 2ĵ − k̂); Cartesian: (x − 2)/1 = (y + 1)/2 = (z − 4)/(−1)
The given line has direction ratios 3, 5, 6. A parallel line has the same direction ratios.
Through (−2, 4, −5): (x + 2)/3 = (y − 4)/5 = (z + 5)/6.
Answer: (x + 2)/3 = (y − 4)/5 = (z + 5)/6
The line passes through (5, −4, 6) and has direction ratios 3, 7, 2.
Answer: r = 5î − 4ĵ + 6k̂ + λ(3î + 7ĵ + 2k̂)
(i) b₁ = 3î + 2ĵ + 6k̂ and b₂ = î + 2ĵ + 2k̂. b₁ · b₂ = 3 + 4 + 12 = 19. |b₁| = 7, |b₂| = 3.
cos θ = 19/(7 × 3) = 19/21.
(ii) b₁ = î − ĵ − 2k̂ and b₂ = 3î − 5ĵ − 4k̂. b₁ · b₂ = 3 + 5 + 8 = 16. |b₁| = √6, |b₂| = √50 = 5√2.
cos θ = 16/(√6 · 5√2) = 16/(10√3) = 8√3/15.
Answer: (i) θ = cos⁻¹(19/21); (ii) θ = cos⁻¹(8√3/15)
(i) Direction ratios (2, 5, −3) and (−1, 8, 4). Dot product = −2 + 40 − 12 = 26. Magnitudes √38 and √81 = 9.
cos θ = 26/(9√38).
(ii) Direction ratios (2, 2, 1) and (4, 1, 8). Dot product = 8 + 2 + 8 = 18. Magnitudes 3 and 9.
cos θ = 18/27 = 2/3.
Answer: (i) θ = cos⁻¹(26/(9√38)); (ii) θ = cos⁻¹(2/3)
Rewrite so that x, y, z have coefficient 1.
First line: (x − 1)/(−3) = (y − 2)/(2p/7) = (z − 3)/2, direction ratios −3, 2p/7, 2.
Second line: (x − 1)/(−3p/7) = (y − 5)/1 = (z − 6)/(−5), direction ratios −3p/7, 1, −5.
Perpendicular: (−3)(−3p/7) + (2p/7)(1) + 2(−5) = 0, so 9p/7 + 2p/7 = 10 and 11p/7 = 10.
Answer: p = 70/11
Direction ratios (7, −5, 1) and (1, 2, 3).
Sum of products: 7(1) + (−5)(2) + 1(3) = 7 − 10 + 3 = 0.
Answer: Shown
a₁ = î + 2ĵ + k̂, b₁ = î − ĵ + k̂, a₂ = 2î − ĵ − k̂, b₂ = 2î + ĵ + 2k̂.
b₁ × b₂ = −3î + 3k̂, with magnitude 3√2.
a₂ − a₁ = î − 3ĵ − 2k̂. (b₁ × b₂) · (a₂ − a₁) = −9.
d = |−9|/3√2.
Answer: 3√2/2
a₁ = (−1, −1, −1), b₁ = 7î − 6ĵ + k̂; a₂ = (3, 5, 7), b₂ = î − 2ĵ + k̂.
b₁ × b₂ = −4î − 6ĵ − 8k̂, with magnitude √(16 + 36 + 64) = 2√29.
a₂ − a₁ = 4î + 6ĵ + 8k̂. (b₁ × b₂) · (a₂ − a₁) = −116.
d = 116/2√29 = 2√29.
Answer: 2√29
b₁ × b₂ = −9î + 3ĵ + 9k̂, with magnitude 3√19.
a₂ − a₁ = 3î + 3ĵ + 3k̂. (b₁ × b₂) · (a₂ − a₁) = 9.
d = 9/3√19 = 3√19/19.
Answer: 3√19/19
Write each line as a point plus a multiple of a direction.
First: r = î − 2ĵ + 3k̂ + t(−î + ĵ − 2k̂). Second: r = î − ĵ − k̂ + s(î + 2ĵ − 2k̂).
b₁ × b₂ = 2î − 4ĵ − 3k̂, with magnitude √29.
a₂ − a₁ = ĵ − 4k̂. (b₁ × b₂) · (a₂ − a₁) = 8.
d = 8/√29 = 8√29/29.
Answer: 8√29/29
Sum of products: a(b − c) + b(c − a) + c(a − b) = ab − ac + bc − ab + ac − bc = 0.
The dot product of the direction ratios is zero, so the lines are perpendicular.
Answer: 90°
The x-axis has direction ratios 1, 0, 0. The line passes through (0, 0, 0).
Answer: r = λî; Cartesian: x/1 = y/0 = z/0 (that is y = 0 and z = 0)
Direction ratios: (−3, 2k, 2) and (3k, 1, −5).
Perpendicular: (−3)(3k) + (2k)(1) + (2)(−5) = −9k + 2k − 10 = 0, so −7k = 10.
Answer: k = −10/7
a₁ = 6î + 2ĵ + 2k̂, b₁ = î − 2ĵ + 2k̂, a₂ = −4î − k̂, b₂ = 3î − 2ĵ − 2k̂.
b₁ × b₂ = 8î + 8ĵ + 4k̂, with magnitude √(64 + 64 + 16) = 12.
a₂ − a₁ = −10î − 2ĵ − 3k̂. (b₁ × b₂) · (a₂ − a₁) = −108.
d = |−108|/12 = 9.
Answer: 9
The two lines have directions b₁ = 3î − 16ĵ + 7k̂ and b₂ = 3î + 8ĵ − 5k̂.
A line perpendicular to both is parallel to b₁ × b₂ = 24î + 36ĵ + 72k̂ = 12(2î + 3ĵ + 6k̂).
Through (1, 2, −4) with direction 2î + 3ĵ + 6k̂:
Answer: r = î + 2ĵ − 4k̂ + λ(2î + 3ĵ + 6k̂)