What is Perpendicular Line?

Use this calculator to compute Perpendicular Line in coordinate geometry.

How do I use this calculator?

Enter the required coordinates or parameters; results and steps update automatically.

Where is Perpendicular Line used?

Coordinate geometry, calculus, physics, and engineering applications.

What units can I use?

Coordinates are unitless numbers; angles in degrees or radians depending on the calculator.

Can I get step-by-step solutions?

Yes. The calculator shows step-by-step derivation when results are displayed.

How do I share my results?

Use the copy or share buttons to export or share your calculation.

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Perpendicular Line

Perpendicular lines satisfy m₁·m₂ = −1. So m_perp = −1/m. Given line with slope m and point (x₀,y₀), perpendicular line: y−y₀ = (−1/m)(x−x₀). Vertical ⟂ horizontal.

Concept Fundamentals

m₁·m₂ = −1

Condition

−1/m

m_perp

m=0 ⟂ x=k

Vertical

Angle between = 90°

90°

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m_perp = −1/m; product m·m_perp = −1. Horizontal ⟂ vertical. Shortest distance from point to line is along perpendicular.

Key quantities

m₁·m₂ = −1

Condition

Key relation

−1/m

m_perp

Key relation

m=0 ⟂ x=k

Vertical

Key relation

Angle between = 90°

90°

Key relation

Ready to run the numbers?

Why: Perpendicular lines form right angles. Essential in geometry (altitudes, medians), construction (right angles), and coordinate geometry. Used for shortest distance from point to line.

How: m_perp = −1/m. For horizontal (m=0), perpendicular is vertical (x=x₀). For vertical, perpendicular is horizontal (y=y₀). Point-slope: y−y₀ = m_perp(x−x₀).

m_perp = −1/m; product m·m_perp = −1.Horizontal ⟂ vertical.

Sources:Khan Academy — Perpendicular LinesWolfram MathWorld — Perpendicular

Run the calculator when you are ready.

Find Perpendicular LineEnter line and point; get perpendicular through point

Original Line (y = mx + b)

Slope (m)

Y-Intercept (b)

Point for Perpendicular Line to Pass Through

x-coordinate

y-coordinate

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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Perpendicular: m₁·m₂ = −1, so m_perp = −1/m.

— Coordinate Geometry

90°

Perpendicular lines meet at 90°.

— Property

Key Takeaways

• Perpendicular lines intersect at a 90° angle
• Slopes are negative reciprocals: $m_1 \\cdot m_2 = -1$ or $m_{\\perp} = -\\frac{1}{m}$
• A horizontal line (m=0) is perpendicular to a vertical line (undefined slope)
• Use point-slope form with the perpendicular slope to find the equation through a point
• The shortest distance from a point to a line is along the perpendicular from the point to the line

Did You Know?

Right Angles

Perpendicular lines form right angles (90°). The product of their slopes is always -1 for non-vertical, non-horizontal lines.

Shortest Distance

The perpendicular from a point to a line gives the shortest distance. Used in geometry, physics, and optimization.

Coordinate Axes

The x-axis and y-axis are perpendicular. This orthogonal system is the foundation of Cartesian coordinates.

Construction

Compass-and-straightedge constructions use perpendicular bisectors to find centers of circles and construct regular polygons.

Vector Dot Product

In vector form, two lines are perpendicular if their direction vectors have dot product zero.

Trigonometry

The tangent of 90° is undefined, matching the fact that a vertical line has undefined slope — perpendicular to horizontal.

Understanding Perpendicular Lines

Two lines are perpendicular if they intersect at a 90° angle. For non-vertical lines with slopes $m_1$ and $m_2$ , perpendicularity means:

m_1 \\cdot m_2 = -1 \\quad \\text{or} \\quad m_2 = -\\frac{1}{m_1}

Given a line $y = mx + b$ and a point $(x_0, y_0)$ , the perpendicular line through that point has slope $m_{\\perp} = -1/m$ and equation:

y - y_0 = m_{\\perp}(x - x_0) \\implies y = m_{\\perp}x + (y_0 - m_{\\perp} x_0)

Expert Tips

Negative Reciprocal

Flip the slope and change the sign. If m = 3, then m⊥ = -1/3. If m = -2/5, then m⊥ = 5/2.

Special Cases

Horizontal (m=0) → perpendicular is vertical (x = c). Vertical → perpendicular is horizontal (y = k).

Verify with Product

Always check: m · m⊥ = -1. If your product is -1, the lines are perpendicular.

Shortest Distance

The perpendicular from a point to a line gives the minimum distance. Used in point-to-line distance formulas.

Frequently Asked Questions

What is the perpendicular slope?

If a line has slope m, the perpendicular slope is -1/m (the negative reciprocal). For example, m = 2 gives m⊥ = -1/2.

What if the original line is horizontal?

A horizontal line has slope 0. The perpendicular line is vertical with equation x = k, where k is the x-coordinate of the point.

What if the original line is vertical?

A vertical line has undefined slope. The perpendicular line is horizontal with equation y = k, where k is the y-coordinate of the point.

Why does m₁·m₂ = -1 work?

Two lines with slopes m₁ and m₂ are perpendicular if the angle between them is 90°. Using trigonometry, tan(θ₁)·tan(θ₂) = -1 when θ₁ + θ₂ = 90°, which gives m₁·m₂ = -1.

Can I use this for 3D?

In 3D, perpendicularity is defined by direction vectors. Two vectors are perpendicular if their dot product is zero.

How is this used in real life?

Perpendicular lines appear in construction (right angles), navigation (orthogonal directions), computer graphics (surface normals), and optimization (gradient descent).

What is the relationship to parallel lines?

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals. They are complementary concepts in coordinate geometry.

How to Use This Calculator

Enter the slope and y-intercept of the original line (y = mx + b).
Enter the point that the perpendicular line must pass through.
Click "Calculate" to get the perpendicular line equation.
Review the visualization showing both lines meeting at 90°.
Check the step-by-step solution for the derivation.
Copy results to share or paste into assignments.

Note: For horizontal (m=0) or vertical lines, the calculator handles the special cases: horizontal → perpendicular is vertical (x = constant), vertical → perpendicular is horizontal (y = constant).

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⬅️Jump in and explore the concept!

Perpendicular Line

Original Line (y = mx + b)

Point for Perpendicular Line to Pass Through

🧮 Fascinating Math Facts

Key Takeaways

Did You Know?

Understanding Perpendicular Lines

Expert Tips

Negative Reciprocal

Special Cases

Verify with Product

Shortest Distance

Frequently Asked Questions

What is the perpendicular slope?

What if the original line is horizontal?

What if the original line is vertical?

Why does m₁·m₂ = -1 work?

Can I use this for 3D?

How is this used in real life?

What is the relationship to parallel lines?

How to Use This Calculator

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We Value Your Privacy

Perpendicular Line

Original Line (y = mx + b)

Point for Perpendicular Line to Pass Through

🧮 Fascinating Math Facts

Key Takeaways

Did You Know?

Understanding Perpendicular Lines

Expert Tips

Negative Reciprocal

Special Cases

Verify with Product

Shortest Distance

Frequently Asked Questions

What is the perpendicular slope?

What if the original line is horizontal?

What if the original line is vertical?

Why does m₁·m₂ = -1 work?

Can I use this for 3D?

How is this used in real life?

What is the relationship to parallel lines?

How to Use This Calculator

Related Mathematics Calculators

Related Calculators

Parallel Line Calculator

Centroid Calculator

Endpoint Calculator

Gradient Calculator

Rotation Calculator

Vector Calculator

We Value Your Privacy