A line in mathematics and geometry is a fundamental concept representing a straight, one-dimensional figure that extends infinitely in both directions. Lines are characterized by having no thickness and being perfectly straight. Here are some important aspects and definitions related to lines:
Key Characteristics of Lines
- Infinite Length: A line extends without end in both directions. It has no beginning or endpoints.
- No Thickness: A line is considered to have no width or depth; it is purely one-dimensional.
- Straightness: A line does not curve or bend. It is the shortest distance between any two points lying on it.
Table of Contents
Straight Lines in 3D space are generally represented in two forms: Cartesian Form and Vector Form. Hence, the angles between any two straight lines in 3D space are also defined in terms of both the forms of the straight lines. Let’s discuss the methods of finding the angle between two straight lines in both forms one by one.
Cartesian Form of Line
L1: (x – x1) / a1 = (y – y1) / b1 = (z – z1) / c1
L2: (x – x2) / a2 = (y – y2) / b2 = (z – z2) / c2
Here L1 & L2 represent the two straight lines passing through the points (x1, y1, z1) and (x2, y2, z2) respectively, in 3D space in Cartesian Form.
- Direction ratios of line L1 are a1, b1, c1 then a vector parallel to L1 is m1 = a1 i + b1 j + c1 k
- Direction ratios of line L2 are a2, b2, c2 then a vector parallel to L2 is m2 = a2 i + b2 j + c2 k
Angle Between Lines in Cartesian Form
Then the angle ∅ between L1 and L2 is given by:
∅ = cos-1{(m1 . m2) / (|m1| × |m2|)}
Solved Examples
Example 1: (x – 1) / 1 = (2y + 3) / 3 = (z + 5) / 2 and (x – 2) / 3 = (y + 1) / -2 = (z – 2) / 0 are the two lines in 3D space then the angle ∅ between them is given by:
Solution:
Given: m1 = 1 i + (3 / 2) j + 2 k and m2 = 3 i – 2 j + 0 k
⇒ |m1| = √(12 + (3/2)2 + 22) = √(29 / 2), and
⇒ |m2| = √(32 + 22 + 02) = √(13)
Thus, ∅ = cos-1{(1×3 + (3/2)×(-2) + (2)×0 ) / ((√(29) / 2) × √(13))}
⇒ ∅ = cos-1{0 / ((√(29) / 2) × √(13))}
⇒ ∅ = cos-1(0)
⇒ ∅ = π / 2
Example 2: Find the angles between the two lines in 3D space whose only direction ratios are given 2, 1, 2 and 2, 3, 1. In the question, equations of the 2 lines are not given, only their DRs are given. So the angle ∅ between the 2 lines is given by:
Solution:
m1 = Vector parallel to the line having DRs 2, 1, 2 = (2 i + j + 2 k)
v|m1| = √(22 + 12 + 22) = √9 = 3
m2 = Vector parallel to the line having DRs 2, 3, 1 = (2 i + 3 j + k)
⇒ |m2| = √(22 + 32 + 12) = √(14)
Thus, ∅ = cos-1{(2×2 + 1×3 + 2×1) / (3 × √(14))}
⇒ ∅ = cos-1{(4 + 3 + 2) / (3 × √(14))}
⇒ ∅ = cos-1{9 / (3 × √(14))}
⇒ ∅ = cos-1(3 / √(14))
Example 3: (x – 1) / 2 = (y – 2) / 1 = (z – 3) / 2 and (x – 2) / 2 = (y – 1) / 2 = (z – 3) / 1 are the two lines in 3D space then the angle ∅ between them is given by:
Solution:
m1 = 2 i + j + 2 k
⇒ |m1| = √(22 + 12 + 22) = √9 = 3
m2 = 2 i + 2 j + k
⇒ |m2| = √(22 + 22 + 12) = √9 = 3
Thus, ∅ = cos-1{(2×2 + 1×2 + 2×1 ) / (3 × 3)}
⇒ ∅ = cos-1{(4 + 2 + 2) / 9}
⇒ ∅ = cos-1(8 / 9)
Vector Form of Line
L1: r = a1 + t . b1
L2: r = a2 + u . b2
Where,
- L1 & L2 represent the two straight lines passing through the points whose position vectors are a1 and a2 respectively in 3D space in Vector Form,
- b1 & b2 are the two vectors parallel to L1 and L2 respectively
- and t & u are the parameters.
Angle Between Lines in Vector Form
Then the angle ∅ between the vectors b⃗b1 and b⃗b2 is equals to the angle between L1 and L2 is given by:
∅ = cos-1{(b1 . b2) / (|b1| × |b2|)}
Solved Examples on Angles Between two Lines in 3D Space
Example 1: r⃗r = (i + j + k) + t × {(-√3 – 1) i + (√3 – 1) j + 4 k} and r⃗r = (i + j + k) + u × (i + j + 2 k) are the two lines in 3D space then the angle ∅ between them is given by:
Solution:
b1 = (-√3 – 1) i + (√3 – 1) j + 4 k
⇒ |b1| = √{(-√3 – 1)2 + (√3 – 1)2 + 42)} = √(24)
b2 = i + j + 2 k
⇒ |b2| = √(12 + 12 + 22) = √6
Thus, ∅ = cos-1{(-√3 – 1)×1 + (√3 – 1)×1 + 4×2 ) / (√(24) × √6)}
⇒ ∅ = cos-1{6 / (√(24) × √6)}
⇒ ∅ = cos-1(½)
⇒ ∅ = π / 3
Example 2: (i + 2 j + 2 k) and (3 i + 2 j + 6 k) are the two vectors parallel to the two lines in 3D space then the angle ∅ between them is given by:
Solution:
b1 = i + 2 j + 2 k
⇒ |b1| = √(12 + 22 + 22)} = √9 = 3
b2 = 3 i + 2 j + 6 k
⇒ |b2| = √(32 + 22 + 62) = √(49) = 7
Thus, ∅ = cos-1{(1×3 + 2×2 + 2×6) / (7 × 3)}
⇒ ∅ = cos-1{(3 + 4 + 12) / 21}
⇒ ∅ = cos-1(19 / 21)
Example 3: r⃗r = (3 i + 5 j + 7 k) + s × {(i + 2 j – 2 k} and r⃗r = (4 i + 3 j + k) + t × (2 i + 4 j – 4 k) are the two lines in 3D space then the angle ∅ between them is given by:
Solution:
b1 = i + 2 j – 2 k
⇒ |b1| = √(12 + 22 + (-2)2)} = √9 = 3
b2 = 2 i + 4 j – 4 k
⇒ |b2| = √(22 + 42 + (-4)2) = √(36) = 6
Thus, ∅ = cos-1{(1×2 + 2×4 + (-2)×(-4)) / (3 × 6)}
⇒ ∅ = cos-1{(2 + 8 + 8) / 18}
⇒ ∅ = cos-1(18 / 18)
⇒ ∅ = cos-1(1) = 0
FAQs on Angles Between Two Lines in 3D Space
What is the angle between two lines in 3D space?
The angle between two lines in 3D space is the smallest angle formed by the two lines when they intersect or if extended to intersect.
What are direction vectors?
Direction vectors are vectors that indicate the direction of a line in 3D space. For a line passing through points (x1, y1, z1) and (x2, y2, z2), the direction vector is (x2−x1, y2−y1, z2−z1).
Can the angle between two lines be more than 90°?
No, the angle between two lines is always taken to be the smallest angle, which is between 0 and 90° inclusive.
What happens if the direction vectors of two lines are perpendicular?
If the direction vectors of two lines are perpendicular, their dot product is zero, which means the angle between the two lines is 90°.
What if the direction vectors of the lines are parallel?
If the direction vectors of two lines are parallel, the angle between the lines is 0°.