Angle of Inclination of Line in Vector Equation

[Cuisine123]

Homework Statement

Find the angle of inclination of each of the following lines.
i) r = (2,-6) + t(3,-4) ii) r = (6,1) + t(5,1)

B) prove that the tangent of the angle of inclination is equal to the slope of the line.

Homework Equations

N/A

The Attempt at a Solution

I know that the angle ø, 0° < ø < 180°, that a line makes with the positive x-axis is called the angle of inclination of the line. However, I don't have any idea how to approach this question.

 

[rock.freak667]

Well your lines are in the xy-plane.
So why don't you draw the lines and find the angle using some trigonometry?

OR you can find the dot product with a unit vector in the direction of the x-axis <1,0>

 

[Cuisine123]

Well your lines are in the xy-plane.
So why don't you draw the lines and find the angle using some trigonometry?

OR you can find the dot product with a unit vector in the direction of the x-axis <1,0>

How do find the solution without having to draw the lines?

 

[rock.freak667]

How do find the solution without having to draw the lines?

the dot product of the direction of the vector line and the unit vector in the direction of the x-axis will give you the angle.

 

[rdl]

I would recommend using the vector equation of a line, r = a + tb, to find the angle of inclination of each line. This equation represents a line passing through a point (a) with a direction vector (b). In this case, we have two points, (2,-6) and (6,1), and two direction vectors, (3,-4) and (5,1). To find the angle of inclination, we can use the dot product formula: cosø = (a⋅b)/(|a||b|), where a and b are the direction vectors. This will give us the cosine of the angle of inclination, which we can then use to find the angle itself by taking the inverse cosine (arccos).

To prove that the tangent of the angle of inclination is equal to the slope of the line, we can use the definition of slope, rise/run, and compare it to the tangent of the angle of inclination, opposite/adjacent. We can show that they are equal by using trigonometric identities, such as sin^2ø + cos^2ø = 1, and substituting in the values we found for cosø and sinø from the dot product formula. This will demonstrate that the tangent of the angle of inclination is indeed equal to the slope of the line.