Angles of Inclination and of Intersection

Angle of Inclination - inc Perpendicular and Parallel Lines

DEFINITION

The Angle of Inclination is the smallest angle (θ) between a line and where it intersects with the positive x-axis. plot of straight line with two points and their delta x and delta y differences shown.

If we recall, for a slope:

m =  
Δ y
Δ x

We also know that the tangent of an angle, tan(θ) = the opposite line to the angle divided by the adjacent line to the angle.

As Δ y = the opposite line of the angle and Δ x = the adjacent line of the angle

We can say that tan(θ) =  
Δ y
Δ x

Therefore,

m = tan(θ)

Parallel Lines

DEFINITION

Two lines will be parallel if each of their angles of inclinations are equal. Remember, the angle of inclination is the angle the line makes with the x-axis in the positive direction

Say we have two lines A and B, with angles of inclination θ1 and θ2

If θ1 = θ2 then the lines are parallel. Similarly, if the slope of A and the slope of B are equal, then they will also be parallel.

We can say that,

Two non-vertical lines are parallel if the slope of each line is equal to the other, m1 = m2.

two parallel lines shown with the angle of intersection with the x-axis shown

Example: Finding equation for a line when knowing a point on the line and the equation of another parallel line

Question: A line of the equation 4x + 2y = 9 runs parallel to a line that has the point (2, 5). Determine the equation to the line with the point (2, 5).

We know that the line of 4x + 2y = 9 runs parallel to another line that has the point (2, 5),

Ask ourselves what can we work with from this statement? What do we know about parallel lines? If two lines are parallel then m1 = m2, meaning the slope of each line is the same.

We want to know the equation to the line that passes through the point (2, 5). We can only work out an equation if we either know; two points on that line or one point on that line and the slope.

We know the slope of the line we want the equation to is equal to the slope of the line 4x + 2y = 9, we can work out the slop of this line!

Rearrange 4x + 2y = 9, to give us something like y =...

Subtract 4x from both sides,

2y = 9 - 4x

Divide both sides by 2,

y =  
9
2
 - 
4x
2
y =   -2x + 
9
2

We know that the slope of a line of the form y = mx + b, is equal to m,

In this case, m = - 2,

Now we have our slope of the line 4x + 2y = 9, we know that this slope is also equal to the slope of the line running parallel that goes through the point (2, 5)

So the slope of the parallel line running through the point (2, 5) is -2, now we have these two values, we can calculate the equation for this line,

Recall, our equation for building an equation of a line is,

y - y1 = m(x - x1)

Where m is the slope, and (x1, y1) is a point on that line,

m = -2

x1 = 2

y1 = 5

So,

y - 5 = -2(x - 2)

y - 5 = -2x + 4

Add 5 to both sides,

y = -2x + 4 + 5

y = -2x + 9

There we have our equation that goes through the point (2, 5) and runs parallel to the line of equation 4x + 2y = 9

Perpendicular Lines

DEFINITION

For two lines that are non-vertical, say A and B, with slopes m1 and m2 respectively, these lines will be perpendicular if,

m1
-1
m2

This is called the negative inverse

two perpendicular lines shown with their angle of intersection shown as a right angle

Example: Finding an equation of a line knowing a point on that line and the equation of a line that is perpendicular

Question: Find the equation to the line that passes through the point (-3, 7) and runs perpendicular to the line of the equation 4y - 3x = 8

Similar to question 8, we use what we know about the relationship between the two lines, as well as what we know from the other line, to allow us to calculate the equation of the line we don't know.

We have two lines that run perpendicular;

One line of equation 4y - 3x = 8

And one line that passes through (-3, 7)

If we find the slope of 4y - 3x = 8, we can find the slope of the line that passes through (-3, 7) as we know that for perpendicular lines the slope of one is equal to the negative inverse of the other, i.e.

m1
-1
m2

Once we have the slope of the line with the point (-3, 7) we can work out the equation for that line.

Start with finding the slope of 4y - 3x = 8,

Add 3x to both sides,

4y = 8 + 3x

Divide both sides by 4,

y =  
3x
4
 + 
8
4

y = (¾)x + 2

So, the slope, m, of 4y - 2x = 8 is (¾)

If we take the negative inverse of this, it will equal the slope of the perpendicular line passing through the point (-3, 7)

This slope becomes -(4/3)

Now we know m = -(4/3) and a point along the line is (-3, 7) we can work out the equation using the formula,

y - y1 = m(x - x1)

with ,

m = -(4/3), x1 = -3, and y1 = 7

y - 7 =  
-4
3
(x - (-3))
y - 7 =  
-4
3
(x) - 
4
3
(3)
y - 7 =  
-4x
3
 - 
4

Add 7 to both sides,

y =  
-4x
3
 - 
4 + 7
y =  
-4x
3
 + 
3

There we have our equation for the line that runs perpendicular to 4y - 3x = 8 and passes through the point (-3, 7)

two perpendicular lines, y = (-4/3)(x) + 3 and 4y - 3x = 8 plotted on an x-y grid

Angles Between Intersecting Lines

DEFINITION

If two non-vertical lines intersect, and that intersection is not perpendicular, then the angle, θ, between those two lines is, where m1 is the slope of the first line (L1) and m2 is the slope of the second line (L2),

θ = arctanopening bracket
m2 - m1
1 + m2m1
closing bracket
two intersecting lines, L1 and L2 shown with the angle of intersection with the x-axis and with each other shown

Proof of equation for angle between lines

Here is the proof:

With θ1 as the angle L1 makes with the x-axis and θ2 is the angle L2 makes with the x-axis,

Trigonometry laws state that in the visual above, θ2 = θ + θ1, therefore,

θ = θ2 - θ1

We also know that tan(θ) = the slope of the line that makes the angle θ with the x-axis. So, applying tan to θ = θ2 - θ1

tan(θ) = tan(θ2 - θ1)

(properties of tan give us)

tan(θ2 - θ1) = 
tan(θ2) - tan(θ1)
1 + tan(θ2)tan(θ1)

And remember, tan of an angle gives us the slope of the line making that angle,

m2 - m1
1 + m2m1

For the arctan or inverse tan of an angle, the result will be between -pi ÷ 2 and pi ÷ 2 or -90° and 90°.

So, with,

tan(θ) = 
m2 - m1
1 + m2m1
θ = arctanopening bracket
m2 - m1
1 + m2m1
closing bracket

Arctan will always give us the smaller of the angles, we have to also account for the larger angle as well, which we get by taking pi - θ or 180° - θ

So, we get our definition,

The smaller angle formed by two non vertical, non perpendicular lines intersecting is,

θ = arctanopening bracket
m2 - m1
1 + m2m1
closing bracket

Example: Finding point of intersection and angle between two lines

Question: For the lines L1(y = x - 2) and L2(y = 2x - 5)two intersecting lines, y = x -2 and y = 2x - 5 with their angle of intersection shown.

a) Find the point of intersection

b) Find the angle of the point of intersection in degrees to the nearest tenth of a decimal place

a) This should be a review of work we have already done, but lets go over it in detail,

The point of intersection is the point where it satisfies both of the equations i.e. it makes both equations true. We find this point by making the equations equal to the same variable (often y) and then equating the other part of the equation and solving it. Here we already have both of our equations equal to y, so we can make the “x-part” of each equation,

y = x - 2

y = 2x - 5

Making them equal,

x - 2 = 2x - 5

Subtract x from both sides,

-2 = 2x - x - 5

-2 = x - 5

Add 5 to both sides,

-2 + 5 = x

x = 3

Now we have our x-value of the intersection point, find the y-value, we do this by subbing the x = 3 into either of the equations (as this is the point that is true for both equations)

Sub x = 3 into y = x - 2,

y = 3 - 2

y = 1

So, our point of intersection is (3,1)

b) now we want to find the angle of intersection between two lines,

We know the equation for an angle between two intersecting lines is,

θ = arctanopening bracket
m2 - m1
1 + m2m1
closing bracket

Where m1 is the slope of the first line and m2 is the slope of the second line,

We can calculate the slope of each line from each equation given,

We know that an equation of the format y = mx + b, that m = slope

So for L1:

y = x -2, m1 = 1

And for L2:

y = 2x - 5, m2 = 5

Now, sub these m1 and m2 values into,

θ = arctanopening bracket
m2 - m1
1 + m2m1
closing bracket
θ = arctanopening bracket
5 - 1
1 + (5)(1)
closing bracket
θ = arctanopening bracket
4
6
closing bracket
= arctan opening bracket
2
3
closing bracket

= 0.5880 radians

Remember, (π)radians = 180°, so:

Radians =  
180°
π

So converting 0.5880 radians into degrees,

(0.5880) x  
180°
π
= 33.689918°

So, θ = 33.7°