Problem 37 - Precalculus Mathematics for Calculus

Question

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 represent the same line? Consider the following two sets of parametric 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 the same.

Step 1: Find Points on Line 1

To find points on Line 1, substitute the given parametric equations with specific values of \( t \). First, set \( t=0 \):- \( x=1-0=1 \)- \( y=3 \times 0=0 \)- \( z=-6+5 \times 0=-6 \)This gives us the point \( (1, 0, -6) \).Next, set \( t=1 \):- \( x=1-1=0 \)- \( y=3 \times 1=3 \)- \( z=-6+5 \times 1=-1 \)This gives us the point \( (0, 3, -1) \).

Step 2: Validate Points on Line 2

Now let's check if the points from Line 1 are also on Line 2.For the point \( (1, 0, -6) \), we solve the equations of Line 2:- Set \( x=-1+2t=1 \) leads to \( t=1 \).- Check \( y=6-6 \times 1=0 \) matches.- Check \( z=4-10 \times 1=-6 \) matches.This confirms \( t=1 \) for Line 2 gives the point \( (1, 0, -6) \).For the point \( (0, 3, -1) \), we solve:- Set \( x=-1+2t=0 \) leads to \( t=0.5 \).- Check \( y=6-6 \times 0.5=3 \) matches.- Check \( z=4-10 \times 0.5=-1 \) matches.This confirms \( t=0.5 \) for Line 2 gives the point \( (0, 3, -1) \).

Step 3: Find and Verify a Point on Line 3

We find a point on Line 3 by setting \( t=0 \):- \( x=4 \times 0 =0 \)- \( y=3-6 \times 0 =3 \)- \( z=-5+2 \times 0 =-5 \)This gives the point \( (0, 3, -5) \).

Step 4: Check if Point is on Line 4

Now check if the point \( (0, 3, -5) \) lies on Line 4:- Set \( x=8-2t=0 \) gives \( t=4 \).- Check \( y=-9+3 \times 4=-9+12=3 \) matches.- Check \( z=6-4=-5+2 \) does not match (-5 expected).The point \( (0, 3, -5) \) is not on Line 4, confirming Lines 3 and 4 are not the same.

Key Concepts

Vector Algebra

Vector algebra is a fundamental tool used to understand various mathematical entities and relationships, especially in physics and engineering. It involves operations with vectors, which are quantities possessing both magnitude and direction, unlike scalars that only have magnitude.

Key operations in vector algebra include:

• Addition and Subtraction: Vectors can be added by summing their components or subtracted by finding the difference between their components.
• Scalar Multiplication: This involves multiplying a vector by a scalar, which affects the magnitude of the vector but not its direction unless the scalar is negative.
• Dot Product: This operation gives a scalar result and is useful in finding the angle between two vectors or determining if they are perpendicular.
• Cross Product: This results in another vector perpendicular to the plane made by the initial vectors and is particularly useful in finding torque or rotational effects.

Understanding these operations is crucial when dealing with parametric equations of lines, as these equations essentially express lines in terms of vector sums and directions.

Line Representation

A line in 3D space can be represented using parametric equations or vector equations. Using parametric equations, the position of points on the line is described as functions of a parameter, usually denoted as \( t \). Each coordinate of the line is given in terms of this parameter, such as \( x = f(t) \), \( y = g(t) \), and \( z = h(t) \).

However, the vector representation offers a more compact and insightful view. A line can be expressed using a vector equation:
\[ \mathbf{r}(t) = \mathbf{r_0} + t\mathbf{v} \]
where \( \mathbf{r}(t) \) is the position vector of a point on the line at parameter \( t \), \( \mathbf{r_0} \) is a fixed point on the line, and \( \mathbf{v} \) is a direction vector parallel to the line.

When comparing different parametric forms, like Lines 1 and 2 in the original exercise, it is important to ensure that the direction vectors are scalar multiples and that there exists a parameter value for which the position vectors of the lines are equal. This assures that the lines are indeed the same.

Coordinate Geometry

Coordinate geometry blends algebra and geometry to study geometric shapes using a coordinate system. By placing points in space using coordinates, we can use algebraic equations to describe and analyze geometric elements like lines and planes.

In the context of lines, coordinate geometry aids in determining relationships among different lines, such as intersection, parallelism, and equivalence as seen in the example. To determine if two lines are equivalent, we look at:

• Direction Vectors: These indicate the line's direction through parametric coefficients, and lines with proportional direction vectors are parallel.
• Common Points: If two lines share at least one point, they might be the same, provided their direction vectors are scalar multiples.

Other aspects such as the angle between lines, line to plane distances, or line intersections further involve coordinate geometry, enriching its utility in 3D space.