**Direction cosines (d.c.'s)** are the cosines of the angles that a line makes with the positive directions of the coordinate axes.
If a directed line L makes angles α, β, γ with the x, y, z axes respectively, then:
**Direction ratios (d.r.'s)** are any three numbers proportional to the direction cosines.
If a, b, c are direction ratios and l, m, n are direction cosines, then:
**Important distinction:**
**Example:** If direction ratios are 1, 2, 2:
For a line passing through P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
**Direction cosines:**
where **PQ = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]**
**Direction ratios (simpler form):** Take a, b, c = (x₂ - x₁), (y₂ - y₁), (z₂ - z₁)
**Example:** Find direction cosines of line through A(2, 3, -1) and B(5, -1, 3):
**Testing collinearity:** Three points A, B, C are collinear if direction ratios of AB are proportional to direction ratios of BC.
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A line in 3D is uniquely determined by:
1. A point on the line and its direction, OR
2. Two points on the line
For a line passing through point A (position vector **a**) parallel to vector **b**:
**Vector equation: r = a + λb** (where λ is a scalar parameter)
**Example:** Line through (1, 2, 3) parallel to vector **i** + 2**j** - **k**:
If point is (x₁, y₁, z₁) and direction ratios are a, b, c:
**x = x₁ + λa**
**y = y₁ + λb**
**z = z₁ + λc**
where λ is the parameter
Eliminating parameter λ from parametric form:
**Standard form: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c**
**If direction cosines l, m, n are given:**
**(x - x₁)/l = (y - y₁)/m = (z - z₁)/n**
**Example:** Find Cartesian form of line through (2, 1, 3) with direction ratios 2, -1, 4:
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The angle θ between two lines is measured as the **acute angle**, so 0 ≤ θ ≤ π/2.
For lines with direction ratios a₁, b₁, c₁ and a₂, b₂, c₂:
**cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / [√(a₁² + b₁² + c₁²) · √(a₂² + b₂² + c₂²)]**
For lines with direction cosines l₁, m₁, n₁ and l₂, m₂, n₂:
**cos θ = |l₁l₂ + m₁m₂ + n₁n₂|** (absolute value because we want acute angle)
For lines **r = a₁ + λb₁** and **r = a₂ + μb₂**:
**cos θ = |b₁ · b₂| / (|b₁| · |b₂|)**
**Two lines are perpendicular (θ = 90°):**
**Two lines are parallel (θ = 0°):**
**Example:** Find angle between lines with direction ratios (1, 2, 2) and (2, 3, -6):
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Two lines that are **not parallel and do not intersect** are called skew lines. The shortest distance is the length of the perpendicular segment joining the two lines.
For lines **r = a₁ + λb₁** and **r = a₂ + μb₂**:
**d = |(b₁ × b₂) · (a₂ - a₁)| / |b₁ × b₂|**
This formula gives:
**Step-by-step calculation:**
1. Find b₁ × b₂ (cross product)
2. Calculate |b₁ × b₂|
3. Find a₂ - a₁
4. Calculate dot product (b₁ × b₂) · (a₂ - a₁)
5. Apply formula: d = |dot product| / |cross product|
**Example:** Lines **r = i + j + λ(i - j + k)** and **r = 2i + j - k + μ(3i - 5j + 2k)**
For lines **(x - x₁)/a₁ = (y - y₁)/b₁ = (z - z₁)/c₁** and **(x - x₂)/a₂ = (y - y₂)/b₂ = (z - z₂)/c₂**:
**d = |[(x₂-x₁)(b₁c₂ - b₂c₁) + (y₂-y₁)(c₁a₂ - c₂a₁) + (z₂-z₁)(a₁b₂ - a₂b₁)]| / √[(b₁c₂ - b₂c₁)² + (c₁a₂ - c₂a₁)² + (a₁b₂ - a₂b₁)²]**
This is complex in Cartesian form; vector form is preferred.
For parallel lines **r = a₁ + λb** and **r = a₂ + μb** (same direction **b**):
**d = |b × (a₂ - a₁)| / |b|**
**Example:** Lines **r = 2i + 4j - k + λ(2i + 3j + 6k)** and **r = 3i + 3j + 5k + μ(2i + 3j + 6k)**
**KEY EXAM POINT:** Always verify if lines are parallel, intersecting, or skew before choosing formula.
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| Concept | Formula |
|---------|---------|
| Direction cosines property | l² + m² + n² = 1 |
| d.c. from d.r. (a,b,c) | l = ±a/√(a²+b²+c²) |
| Line equation (vector) | **r** = **a** + λ**b** |
| Line equation (Cartesian) | (x-x₁)/a = (y-y₁)/b = (z-z₁)/c |
| Angle between lines (cosine) | cos θ = \|a₁a₂ + b₁b₂ + c₁c₂\| / [√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)] |
| Perpendicular lines | a₁a₂ + b₁b₂ + c₁c₂ = 0 |
| Parallel lines | a₁/a₂ = b₁/b₂ = c₁/c₂ |
| Distance between skew lines | d = \|(b₁ × b₂) · (a₂ - a₁)\| / \|b₁ × b₂\| |
| Distance between parallel lines | d = \|b × (a₂ - a₁)\| / \|b\| |
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1. **Forgetting absolute value in angle formula:** The dot product can be negative, but angle must be acute, so use absolute value
2. **Confusing direction cosines with ratios:** d.c.'s satisfy l² + m² + n² = 1; d.r.'s do not
3. **Not checking if lines are parallel before using skew distance formula:** Parallel lines require different approach
4. **Sign errors in vector cross product:** Carefully expand 3×3 determinant using cofactors
5. **Parametric vs Cartesian form:** Parametric has parameter λ; Cartesian eliminates it
6. **Not using magnitude correctly:** When converting direction ratios to cosines, ALWAYS take positive square root first, then ± signs
Q1. If a line makes angles 90°, 60°, and 30° with the positive x, y, and z axes respectively, what are its direction cosines?
Answer: A — Direction cosines are cos 90° = 0, cos 60° = 1/2, and cos 30° = √3/2.
Q2. A line has direction ratios 1, 2, 2. Which of the following are its direction cosines?
Answer: A — √(1² + 2² + 2²) = √9 = 3, so direction cosines are 1/3, 2/3, 2/3, and 1/9 + 4/9 + 4/9 = 1 ✓
Q3. What are the direction cosines of the y-axis?
Answer: B — The y-axis makes angles 90°, 0°, 90° with x, y, z axes, giving direction cosines 0, 1, 0.
Q4. Find the direction ratios of the line passing through points A(1, 2, 3) and B(4, 5, 6).
Answer: A — Direction ratios = (4−1, 5−2, 6−3) = (3, 3, 3); note (B) is a simplified proportional form, but (A) is exact.
Q5. If l, m, n are direction cosines of a line, which statement is always true?
Answer: B — The fundamental property of direction cosines is that the sum of their squares equals 1.
Q6. Show that points A(2, 3, −4), B(1, −2, 3), and C(3, 8, −11) are collinear. Which condition must be satisfied?
Answer: B — Collinearity requires proportional direction ratios: AB has ratios (−1, −5, 7) and BC has (2, 10, −14) = −2 × (−1, −5, 7).
Q7. Which of the following is NOT a correct statement about direction cosines and direction ratios?
Answer: D — Direction cosines and ratios have the same values only by coincidence; in general, ratios must be normalized by √(a² + b² + c²) to get cosines.
Q8. If direction cosines of a line are l = 2/√29, m = 3/√29, n = ?. What is n?
Answer: C — Using l² + m² + n² = 1: (4 + 9)/29 + n² = 1 gives n² = 16/29, so n = ±4/√29.
Q9. Two lines have direction ratios (1, 2, 3) and (2, 4, 6) respectively. What is their relationship?
Answer: B — The second set (2, 4, 6) = 2 × (1, 2, 3), so direction ratios are proportional, meaning the lines are parallel.
Q10. The Cartesian equation of a line passing through (2, 3, 4) with direction ratios 1, −1, 2 is (x − 2)/1 = (y − 3)/(−1) = (z − 4)/2. If a point (x, y, z) lies on this line with parameter value λ = 3, find the coordinates of the point.
Answer: A — From the Cartesian form: x = 2 + λ(1) = 2 + 3 = 5, y = 3 + λ(−1) = 3 − 3 = 0, z = 4 + λ(2) = 4 + 6 = 10.
What is the fundamental property that all direction cosines must satisfy?
The sum of squares of direction cosines equals one: l² + m² + n² = 1.
How do you convert direction ratios a, b, c into direction cosines l, m, n?
Divide each ratio by the square root of the sum of their squares: l = a/√(a² + b² + c²), and similarly for m and n.
Find the direction cosines of the x-axis.
The x-axis makes angles 0°, 90°, 90° with x, y, z axes respectively, so its direction cosines are 1, 0, 0.
What are the direction ratios of the line joining points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂)?
The direction ratios are x₂ − x₁, y₂ − y₁, z₂ − z₁ (or their negative multiples).
If a line passes through point A with position vector **a** and is parallel to vector **b**, what is its vector equation?
The vector equation is **r** = **a** + λ**b**, where λ is a real parameter.
Write the Cartesian equation of a line through point (x₁, y₁, z₁) with direction ratios a, b, c.
The Cartesian form is (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
How do you check if three points A, B, C are collinear using direction ratios?
Find direction ratios of AB and BC; if they are proportional, the points are collinear.
State the formula for the direction cosines of a line joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).
Direction cosines are (x₂ − x₁)/PQ, (y₂ − y₁)/PQ, (z₂ − z₁)/PQ, where PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
Why does a given line in space have two sets of direction cosines?
A line can be extended in two opposite directions, and the direction cosines change sign when the direction reverses.
If direction ratios of a line are 2, −1, −2, what are its direction cosines?
First find √(4 + 1 + 4) = 3, then direction cosines are ±(2/3, −1/3, −2/3).
If a line makes equal angles with the coordinate axes, find its direction cosines. [2 marks]
Let each angle be θ; then l = m = n = cos θ. Use the constraint l² + m² + n² = 1 to solve for cos θ, giving l = m = n = ±1/√3.
Derive the formula for direction cosines of a line passing through two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). Hence, find the direction cosines of the line joining (−2, 4, −5) and (1, 2, 3). [5 marks]
The direction cosines are proportional to (x₂ − x₁, y₂ − y₁, z₂ − z₁); normalize by dividing by PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. Here PQ = √77, and direction cosines are 3/√77, −2/√77, 8/√77.
Using direction ratios, prove that the points A(3, 5, −4), B(−1, 1, 2), and C(−5, −5, −2) are collinear. Also, write the vector equation of the line passing through these points. [6 marks]
Calculate direction ratios of AB and BC; verify they are proportional (both should be multiples of (−4, −4, 6) or equivalent). Write vector equation as **r** = **a** + λ**b** where **a** is position vector of A and **b** is the common direction vector; alternatively verify using parametric form that all three points satisfy the same linear relationship.
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