Mathwords logoMathwords

Orthogonal Lines — Definition, Formula & Examples

Orthogonal lines are two lines that intersect at a right angle (90°). The term 'orthogonal' is simply a more formal or technical synonym for 'perpendicular.'

Two lines in a plane are orthogonal if and only if they meet at an angle of 90°90°. In coordinate geometry, two non-vertical lines with slopes m1m_1 and m2m_2 are orthogonal when m1⋅m2=−1m_1 \cdot m_2 = -1.

Key Formula

m1⋅m2=−1m_1 \cdot m_2 = -1
Where:
  • m1m_1 = Slope of the first line
  • m2m_2 = Slope of the second line

Worked Example

Problem: Determine whether the lines y = 3x + 2 and y = -⅓x + 5 are orthogonal.
Identify the slopes: Read the slopes directly from slope-intercept form.
m1=3,m2=−13m_1 = 3, \quad m_2 = -\tfrac{1}{3}
Multiply the slopes: Check whether their product equals −1-1.
3×(−13)=−13 \times \left(-\tfrac{1}{3}\right) = -1
Conclude: Since the product is −1-1, the lines are orthogonal.
Answer: Yes, the two lines are orthogonal (perpendicular) because the product of their slopes is −1-1.

Why It Matters

The word 'orthogonal' appears frequently in physics, engineering, and linear algebra, where it extends beyond simple lines to describe vectors and even functions that are 'at right angles' in a generalized sense. Recognizing it as equivalent to 'perpendicular' prevents confusion when you encounter it in these more advanced contexts.

Common Mistakes

Mistake: Thinking orthogonal lines must have slopes that are simply negatives of each other (e.g., m1=2m_1 = 2 and m2=−2m_2 = -2).
Correction: The slopes must be negative reciprocals, not just negatives. For m1=2m_1 = 2, the orthogonal slope is m2=−12m_2 = -\tfrac{1}{2}, since 2×(−12)=−12 \times (-\tfrac{1}{2}) = -1.