Problem 37 - Algebra and Trigonometry

Question

Same Line: Different Parametric Equations Every line can be described by infinitely many different sets of parametric equations, since any point on the line and any vector parallel to the line can be used to construct the equations. But how can we tell whether two sets of parametric equations rep- resent the same line? Consider the following two sets of para- metric equations: Line \(1 : \quad x=1-t, \quad y=3 t, \quad z=-6+5 t\) Line \(2 : \quad x=-1+2 t, \quad y=6-6 t, \quad z=4-10 t\) (a) Find two points that lie on Line 1 by setting \(t=0\) and \(t=1\) in its parametric equations. Then show that these points also lie on Line 2 by finding two values of the parameter that give these points when substituted into the parametric equations for Line 2 . (b) Show that the following two lines are not the same by finding a point on Line 3 and then showing that it does not lie on Line \(4 .\) Line \(3 : \quad x=4 t, \quad y=3-6 t, \quad z=-5+2 t\) Line \(4 : \quad x=8-2 t, \quad y=-9+3 t, \quad z=6-t\)

Step-by-Step Solution

Answer

Lines 1 and 2 are the same; Lines 3 and 4 are not.

Step 1: Find points on Line 1

For Line 1, substitute \( t=0 \) into the parametric equations: \( x = 1-0 = 1 \), \( y = 3(0) = 0 \), \( z = -6+5(0) = -6 \). This gives point \((1, 0, -6)\). Now, substitute \( t=1 \): \( x = 1-1 = 0 \), \( y = 3(1) = 3 \), \( z = -6+5(1) = -1 \). This gives point \((0, 3, -1)\).

Step 2: Check if points are on Line 2

For Line 2, find a parameter value \( t \) such that \( x = 1 \), \( y = 0 \), \( z = -6 \). Solving \( -1+2t=1 \) gives \( t = 1 \), \( 6-6t=0 \) gives \( t = 1 \), \( 4-10t=-6 \) gives \( t = 1 \). Thus, \((1, 0, -6)\) lies on Line 2. For \((0, 3, -1)\), solve \( -1+2t=0 \), \( 6-6t=3 \), \( 4-10t=-1 \), all giving \( t = 0.5 \). So, both points are on Line 2.

Step 3: Choose a point on Line 3

For Line 3, substitute \( t=0 \): \( x=4(0)=0 \), \( y=3-6(0)=3 \), \( z=-5+2(0)=-5 \). This results in the point \((0, 3, -5)\).

Step 4: Verify point does not lie on Line 4

Check if \((0, 3, -5)\) can be represented by Line 4. Solving \( 8-2t=0 \) gives \( t=4 \), \( -9+3t=3 \) gives \( t=4 \), but solving \( 6-t=-5 \) gives \( t=11 \). These inconsistent solutions show that \((0, 3, -5)\) is not on Line 4.

Key Concepts

Line Representation

Representing a line using parametric equations involves capturing the infinite set of points lying on that line. These equations typically use a parameter, often denoted as \( t \), to express the coordinates along the line as functions of \( t \).
The general form is:

• \( x = x_0 + at \)
• \( y = y_0 + bt \)
• \( z = z_0 + ct \)

where \((x_0, y_0, z_0)\) is a point on the line, and \( \langle a, b, c \rangle \) is a vector parallel to the line.
This way, as \( t \) changes, different points on the line are generated.

Understanding Parameters

The parameter \( t \) often represents time or progression along the line.
By substituting various values of \( t \), you can find specific points on the line.
These parametric equations are highly adaptable. Changing the point or vector can produce different equations but still represent the same line.

Points on a Line

Finding points on a line defined by parametric equations involves selecting values for the parameter \( t \). Each choice of \( t \) generates a unique point on the line. For instance, in the original exercise, selecting \( t = 0 \) and \( t = 1 \) on Line 1 results in the points \((1, 0, -6)\) and \((0, 3, -1)\) respectively.

Verifying Points on a Line

To confirm that a point lies on a line, substitute the coordinates into the parametric equations. Solve these equations for \( t \), and if a consistent solution exists, the point is on that line. In the exercise, the point \((1, 0, -6)\) was verified as lying on Line 2 by finding \( t = 1 \) across all equations. This verification process shows that the point is indeed part of the same geometric line.
This method can also identify discrepancies which reveal when a point does not lie on a line, as demonstrated with Line 3 and Line 4.

Vector Parallel to a Line

The vector \( \langle a, b, c \rangle \) in parametric equations \( (x = x_0 + at, y = y_0 + bt, z = z_0 + ct) \) is crucial as it indicates direction and orientation.
This vector determines how the coordinates change as the parameter \( t \) varies, defining the "path" along the line.

Properties of Parallel Vectors

Vectors parallel to a line do not have to start from the line itself.
What matters is maintaining the directional consistency that the vector provides.
Different vectors can be parallel to the same line, meaning they could share the same or proportional components. This is why varying vectors can still represent the same geometric line when used with compatible parameter equations.Recognizing these parallel vectors in two different sets of parametric equations can reveal if they describe the same line, alongside tests on points generated by parameter values.