• Engineering and Architecture
  • Management and Business Administration
  • Medicine and Allied Sciences
  • Law
  • Animation and Design
  • Media, Mass Communication and Journalism
  • Finance & Accounts
  • Computer Application and IT
  • Pharmacy
  • Hospitality and Tourism
  • Competition
  • School
  • Study Abroad
  • Arts, Commerce & Sciences
  • Learn
  • Online Courses and Certifications
Angle Between Two Lines

Angle Between Two Lines

Team Careers360Updated on 02 Jul 2025, 05:18 PM IST

When two straight lines intersect, they form two sets of angles. The intersection results in two acute angles and two obtuse angles. The absolute value of angles is determined by the slopes of intersecting lines. The angle between two lines represents the degree of inclination between the two lines and the inclination of two lines is advantageous to comprehend how the lines are associated with each other.

This Story also Contains

  1. The Angle Between Two Lines In Two-Dimensional Space.
  2. The Angle Between Two Lines In Three-Dimensional Space
  3. Vertically Opposite Angles And Adjacent Angles
  4. Examples

The Angle Between Two Lines In Two-Dimensional Space.

The angle between two lines can be computed either by knowing the slope of two lines or by knowing the equations of the two lines. Generally, the acute angle between the lines is calculated.by following methods

  • The angle between two lines when the slope of the lines is known.

To calculate the angle between two lines when the slope of the lines is known, the trigonometric tangent function is used. Take into consideration two lines that have slopes of m1 and m2, respectively. Using the tangent function formula, it is possible to determine the acute angle between the lines. The acute angle \theta 1706462145558 between the lines is given by:

\tan\theta=\frac{m1-m2}{1+m1\cdot m2}

1706462143679

  • The angle between two lines when the equations of the lines are known.

JEE Main Highest Scoring Chapters & Topics
Focus on high-weightage topics with this eBook and prepare smarter. Gain accuracy, speed, and a better chance at scoring higher.
Download E-book

Let the equations of the lines be given by,

l_1=a_1x+b_1y+c_1\\

and \\

l_2=a_2x+b_2y+c_2

1706462145088

then the tangent of the acute angle between the lines can be computed by the following formula.

\tan\theta=\frac{a_2b_1-a_1b_2}{a_1a_2+b_1b_2}

1706462144522

  • If the lines are parallel, the angle between them is zero.

This gives, \tan\theta=0

1706462144685

\frac{m1-m2}{1+m1\cdot m2}=0

1706462143963

m1-m2 = 0,

And hence,

m1=m2

So, this can be concluded that the slope of two parallel lines is the same.

  • If the lines are perpendicular, the angle between them is 90^{\circ} 1706462144285.

This gives, \frac{1}{tan\theta}=0

1706462145389

\frac{1+m1\cdot m2}{m1-m2}=0

1706462145284

1+m1\cdot m2=0

1706462145158

m1\cdot m2=-1

1706462143398

So, the product of slopes of perpendicular lines is equal to -1.

  • Consider a pair of straight lines having the following equation:

ax^2 + 2hxy + by^2 = 0

1706462145492

then the angle between the pair of straight lines is given by

\tan\theta=\frac{2\sqrt{h^2-ab}}{a+b}

1706462144058

Commonly Asked Questions

Q: What's the significance of the angle between lines in real-world applications?
A:
The angle between lines is crucial in various fields like physics (for calculating vector components), engineering (for structural analysis), computer graphics (for 3D modeling), and navigation (for determining directions).
Q: What's the difference between direction ratios and direction cosines?
A:
Direction ratios are any set of numbers proportional to the components of the direction vector. Direction cosines are the cosines of the angles the line makes with the positive x, y, and z axes. Direction cosines are normalized direction ratios.
Q: How do you find the direction vector of a line given two points on the line?
A:
To find the direction vector of a line passing through points P(x1, y1, z1) and Q(x2, y2, z2), subtract the coordinates:
Q: Can you have an obtuse angle between two lines in 3D space?
A:
No, you can't have an obtuse angle between two lines in 3D space. The angle between lines is always defined as the smaller of the two angles formed, which is always acute (0° to 90°) or right (90°).
Q: How does the concept of angle between lines extend to higher dimensions?
A:
In higher dimensions, the angle between lines is still defined using the dot product of their direction vectors. The formula cos θ = |a · b| / (|a| |b|) remains valid, with vectors having more components.

The Angle Between Two Lines In Three-Dimensional Space

The calculation of the angle between two lines in a three-dimensional space is similar to that of in two-dimensional space.

  • Consider two lines with the following equations:

r=a_1+\lambda b_1 \\ and \\ r=a_2+\lambda b_2

1706462144215

the angle between these lines is given by the formula:

\cos\theta=\frac{b_1b_2}{|b_1||b_2|}

1706462144134

  • Consider two lines having direction ratios (a_1,b_1,c_1)\: and\: (a_2,b_2,c_2)

1706462145661

Then the angle between the lines is given by the following formula.

\cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}

1706462145731

  • Consider two lines having the direction cosines as l_1, m_1, n_1 \: and \: l_2,m_2,n_2

then the angle between the lines is calculated by using the following formula.

\cos\theta=|l_1l_2+m_1m_2+n_1n_2|

1706462145866

Commonly Asked Questions

Q: What is meant by the angle between two lines in 3D space?
A:
The angle between two lines in 3D space is the smallest angle formed when the lines are brought together to intersect at a point. It's the angle between the direction vectors of the two lines, measured in the plane containing both lines.
Q: How is the angle between two lines different in 3D compared to 2D?
A:
In 3D, lines can be skew (not intersecting and not parallel), which isn't possible in 2D. The angle between 3D lines is defined even when they don't intersect, using their direction vectors. In 2D, lines always intersect unless parallel.
Q: What is the formula for calculating the angle between two lines in 3D?
A:
The angle θ between two lines with direction vectors a = (a1, a2, a3) and b = (b1, b2, b3) is given by:
Q: Why do we use the absolute value in the angle formula?
A:
The absolute value is used because we're interested in the smallest angle between the lines, which is always acute (0° to 90°). Without the absolute value, we might get the supplementary angle (90° to 180°).
Q: Can the angle between two lines in 3D be greater than 90°?
A:
No, the angle between two lines in 3D is always defined as the smaller of the two angles formed, so it's always between 0° and 90°. The larger angle would be the supplement of this angle.

Vertically Opposite Angles And Adjacent Angles

When two lines intersect, a pair of vertically opposite angles and a pair of adjacent angles are formed.

Pair of vertically opposite angles: (1, 3) and (2, 4)

Pair of adjacent angles: (1, 2), (2, 3), (3, 4) and (4, 1)

  • The measure of vertically opposite angles is equal.

\angle 1 = \angle 3\\

\angle 2 = \angle 4

1706462146126

  • The sum of two adjacent angles is equal to 180 degrees.

\angle 1 +\angle 2=180 ^{\circ}\\

\angle 2 +\angle 3=180 ^{\circ}\\

\angle 3 +\angle 4=180 ^{\circ}\\

\angle 4 +\angle 1=180 ^{\circ}

1706462146219

Commonly Asked Questions

Q: What does it mean if the angle between two lines is 0°?
A:
If the angle between two lines is 0°, it means the lines are parallel or coincident. Their direction vectors point in the same or exactly opposite directions.
Q: What does it mean if the angle between two lines is 90°?
A:
If the angle between two lines is 90°, it means the lines are perpendicular. Their direction vectors are orthogonal (at right angles) to each other.
Q: How can you tell if two lines are perpendicular without calculating the angle?
A:
Two lines are perpendicular if their direction vectors are orthogonal. This occurs when their dot product equals zero. So, if a · b = 0, where a and b are the direction vectors, the lines are perpendicular.
Q: What's the geometric interpretation of the dot product in relation to the angle?
A:
Geometrically, the dot product a · b = |a| |b| cos θ, where θ is the angle between vectors a and b. This relationship forms the basis for calculating the angle between lines using their direction vectors.
Q: How does the angle between lines relate to the dot product of their direction vectors?
A:
The cosine of the angle between two lines is equal to the dot product of their unit direction vectors. For non-unit vectors, we divide the dot product by the product of their magnitudes: cos θ = (a · b) / (|a| |b|)

Examples

  1. Find the measure of the acute angle between the two straight lines having slopes m1=5 and m2=4.

Using the formula, \tan\theta=\frac{m1-m2}{1+m1\cdot m2}

\tan\theta=\frac{5-4}{1+5\cdot 4}

\tan\theta=\frac{1}{21}\\

\theta=\tan^{-1}\frac{1}{21}=0.047^{\circ}

1706462143802

1706462144758

  1. Find the measure of the acute angle between the two straight lines whose equations are x+2y+3=0 and 2x+4y+9=0.

Using the formula, \tan\theta=\frac{a_2b_1-a_1b_2}{a_1a_2+b_1b_2}

1706462144374

\tan\theta=\frac{2\cdot2-1\cdot4}{1\cdot2+2\cdot4}\\

\tan\theta=\frac{0}{10}=0\\

\theta=0^{\circ}

1706462144844

Frequently Asked Questions (FAQs)

Q: How do you find the angle between two lines given their intercepts on the coordinate planes?
A:
If lines have intercepts (a,0,0), (0,b,0), (0,0,c) and (d,0,0), (0,e,0), (0,0,f), their direction vectors are (1/a,1/b,1/c) and (1/d,1/e,1/f). Use these in the standard angle formula: cos θ = |a · b| / (|a| |b|)
Q: How do you find the line that bisects the angle between two given lines?
A:
To find the angle bisector, add the unit direction vectors of the two lines. The resulting vector, when normalized, gives the direction of the angle bisector.
Q: What's the significance of the sine of the angle between two lines?
A:
The sine of the angle between two lines represents the ratio of the magnitude of the cross product of their direction vectors to the product of their magnitudes: sin θ = |a × b| / (|a| |b|)
Q: How does the angle between two lines relate to the concept of orthogonality in linear algebra?
A:
Two lines are orthogonal if and only if the angle between them is 90°. In linear algebra, this corresponds to their direction vectors having a dot product of zero.
Q: How do you find the angle between a line and its projection on a plane?
A:
The angle between a line and its projection on a plane is the complement of the angle between the line and the normal to the plane. If φ is the angle between the line and the plane normal, then 90° - φ is the angle with the projection.
Q: What's the relationship between the angle of two lines and the angle between planes perpendicular to these lines?
A:
The angle between two lines is the same as the angle between planes perpendicular to these lines. This is because the normal vectors of these planes are parallel to the direction vectors of the lines.
Q: How does the concept of angle between lines extend to non-Euclidean geometries?
A:
In non-Euclidean geometries, the concept of angle between lines is more complex. For example, in hyperbolic geometry, the angle between lines is defined using the hyperbolic functions instead of trigonometric functions.
Q: What's the connection between the angle of two lines and the concept of work in physics?
A:
In physics, the work done by a force F along a displacement s is W = F · s = |F| |s| cos θ, where θ is the angle between the force vector and the displacement vector. This uses the same principle as calculating the angle between lines.
Q: How do you find the angle between two lines given their equations in general form?
A:
To find the angle between lines Ax + By + Cz + D = 0 and Ex + Fy + Gz + H = 0, use their normal vectors (A,B,C) and (E,F,G) in the formula: cos θ = |AE + BF + CG| / √((A²+B²+C²)(E²+F²+G²))
Q: How does the angle between two lines affect the area of the parallelogram formed by their direction vectors?
A:
The area of the parallelogram formed by two vectors a and b is |a × b| = |a| |b| sin θ, where θ is the angle between the vectors. As θ approaches 0° or 180°, the area approaches 0.
Articles

News and Notifications

Latest updates