9.5 The Distance from a Point to a Line in R2
Chapter
Chapter 9
Section
9.5
Lectures 4 Videos  Shortest Distance from a Point to 2D scalar equation of the line Shortest Distance between Two Parallel Lines  Shortest Distance between a Point and Line in 3D example
Solutions 27 Videos

Determine the distance from P(-4, 5) to each of the following lines:

3x + 4y - 5 = 0

0.43mins
Q1a

Determine the distance from P(-4, 5) to each of the following lines:

5x -12y + 24 = 0

0.43mins
Q1b

Determine the distance from P(-4, 5) to each of the following lines:

9x - 40y = 0

0.40mins
Q1c

Determine the distance between the following parallel lines:

2x-y+1=0,2x-y+6=0

1.15mins
Q2a

Determine the distance between the following parallel lines:

b) 7x - 24y + 168 =0, 7x - 24y - 336 = 0

2.11mins
Q2b

Determine the distance from R(-2, 3) to each of the following lines:

\vec{r}=(-1,2)+s(3,4), s \in \mathbf{R}

2.29mins
Q3a

Determine the distance from R(-2, 3) to each of the following lines:

\vec{r}=(1,0)+t(5,12), t \in \mathbf{R}

1.29mins
Q3b

Determine the distance from R(-2, 3) to each of the following lines:

\vec{r} = (1, 3) + p(7, -24), p \in \mathbb{R}

1.22mins
Q3c

Find the distance between the following lines:

a) The formula for the distance from a point to a line is d=\displaystyle{\frac{|Ax_0+by_0+C|}{\sqrt{A^2+B^2}}}. Show that this formula can be modified so the distance from the origin, O(0,0), to the line Ax+By+c=0 is given by the formula d=\displaystyle{\frac{|C|}{\sqrt{A^2+B^2}}}.

0.35mins
Q4a

Find the distance between the following lines:

Determine the distance between L_1:3x-4y-12=0 and L_2:3x-4y+12=0 by first finding the distance from the origin to L_1 and then finding the distance from the origin to L_2.

0.46mins
Q4b

Find the distance between the following lines:

b) Determine the distance between L_1:3x-4y-12=0 and L_2:3x-4y+12=0 by first finding the distance from the origin to L_1 and then finding the distance from the origin to L_2.

c) Find the distance between the two lines directly by first determining a point on one of the lines and then using the distance formula. How does this answer compare with the answer you found in part b.?

0.57mins
Q4c

Calculate the distance between the following lines:

\vec{r}=(-2,1)+s(3,4):s \in \mathbb{R}:\vec{r}=(1,0)+t(3,4), t \in \mathbb{R}

1.10mins
Q5a

Calculate the distance between the following lines:

b) \displaystyle{\frac{x-1}{4}}=\displaystyle{\frac{y}{-3}}, \displaystyle{\frac{x}{4}}=\displaystyle{\frac{y+1}{-3}}

0.57mins
Q5b

Calculate the distance between the following lines:

2x-3y+1=0,2x-3y-3=0

0.46mins
Q5c

Calculate the distance between the following lines:

5x+12y=120, 5x+12y+120=0

0.41mins
Q5d

Calculate the distance between point P and the given line.

P(1,2,-1); \vec{r}=(1,0,0)+s(2,-1,2),s \in \mathbf{R}

3.10mins
Q6a

Calculate the distance between point P and the given line.

P(0,-1,0); \vec{r}=(2,1,0)+t(-4,5,20), t \in \mathbf{R}

1.37mins
Q6b

Calculate the distance between point P and the given line.

P(2,3,1); \vec{r}=p(12,-3,4), p \in \mathbf{R}

1.33mins
Q6c

Calculate the distance between the following parallel lines.

\vec{r}=(1,1,0)+s(2,1,2),s \in \mathbf{R}; \vec{r}=(-1,1,2)+t(2,1,2), t \in \mathbf{R}

1.56mins
Q7a

Calculate the distance between the following parallel lines.

\vec{r} = (3,1,-2) + m(1,1,3), m(1,1,3), m \in \mathbf{R}; \vec{r}=(1,0,1)+n(1,1,3), n \in \mathbf{R}

1.34mins
Q7b

a) Determine the coordinates of the point on the line \vec{r}=(1,-1,2)+s(1,3,-1),s \in \mathbf{R}, that produces the shortest distance between the line and a point with coordinates (2,1,3).

3.58mins
Q8a

a) Determine the coordinates of the point on the line \vec{r}=(1,-1,2)+s(1,3,-1),s \in \mathbf{R}, that produces the shortest distance between the line and a point with coordinates (2,1,3).

b) What is the distance between the given point and the line?

1.05mins
Q8b

Two planes with equations x -y + 2z = 2 and x + y - z = -2 intersect along line L. Determine the distance from P(-1, 2, -1) to L, and determine the coordinates of the point on L that gives this minimal distance.

5.47mins
Q9

The point A(2, 4, -5) is reflected in the line with equation \vec{r} = (0, 0, 1) + s(4, 2, 1), s \in \mathbb{R}, to give the point A'. Determine the coordinates of A'.

5.45mins
Q10

A rectangular box with an open top, measuring 2 by 2 by 3, is constructed. Its vertices are labelled as shown. Determine the distance from A to the line segment HB.

1.36mins
Q11a

A rectangular box with an open top, measuring 2 by 2 by 3, is constructed. Its vertices are labelled as shown. b) What other vertices on the box will give the same distance to HB as the distance you found in part a.? c) Determine the area of the \triangle AHB.