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Wize High School Grade 12 Calculus Textbook > Distance Between Lines & Planes

Distance from a Point to a Line in R2R^2

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Distance From a Point to a Line in R2

If Ax+By+C=0Ax+By+C=0 is a line in R2R^2, then the shortest distance between a point P(x1,  y1)P\left(x_1,\ \ y_1\right) and this line is d=∣Ax1+By1+C∣A2+B2\displaystyle d=\frac{\left|Ax_1+By_1+C\right|}{\sqrt{A^2+B^2}}

Example
Find the distance between the point (1, −3)\left(1,\ -3\right) and the line 2x−y+1=02x-y+1=0

d=∣2(1)−(−3)+1∣22+(−1)2=∣2+3+1∣4+1=65d=\frac{\left|2\left(1\right)-\left(-3\right)+1\right|}{\sqrt{2^2+\left(-1\right)^2}}=\frac{\left|2+3+1\right|}{\sqrt{4+1}}=\frac{6}{\sqrt{5}}

Practice: Distance From a Point to a Line in R2

Find the distance between the point P(1,−2)P\left(1,-2\right) and the following lines:
a) 3x−2y−5=03x-2y-5=0
b) r⃗=[0, −1]+t[1, 3]\vec{r}=\left[0,\ -1\right]+t\left[1,\ 3\right]

Practice: Distance Between 2 Parallel Lines in R2

Calculate the distance between the two lines 2x−6y+6=02x-6y+6=0 and x−3y−8=0x-3y-8=0.