Line Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Line

\ell_1

y = 3x + 1

. Line

\ell_2

y = -\frac{1}{3}x + 4

. Are these lines perpendicular? Justify your answer.

1
Step 1: Extract slopes: $m_1 = 3$ , $m_2 = -\tfrac{1}{3}$ .
2
Step 2: Two lines are perpendicular if and only if $m_1 \times m_2 = -1$ .
3
Step 3: $3 \times \left(-\tfrac{1}{3}\right) = -1$ . The condition is satisfied.

Answer

Yes,

\ell_1 \perp \ell_2

because

m_1 \cdot m_2 = -1

Perpendicular lines meet at right angles. Their slopes are negative reciprocals of each other — one is

m

and the other is

-1/m

. This is because rotating a direction vector 90° negates one component and swaps the roles of rise and run.

About Line

A perfectly straight path extending infinitely in both directions through two distinct points, with no thickness.

Write the equation of a line with slope [formula] and y-intercept [formula].

Find the slope of the line passing through points [formula] and [formula].

What is the slope of a horizontal line? What is the slope of a vertical line?