Problem 19 - APEX Calculus

Question

Determine if the described lines are the same line, parallel lines, intersecting or skew lines. If intersecting, give the point of intersection. $$ \ell_{1}=\left\\{\begin{array}{l} x=1+2 t \\ y=3-2 t \\ z=t \end{array}\right. \text { and } \ell_{2}=\left\\{\begin{array}{l} x=3-t \\ y=3+5 t \\ z=2+7 t \end{array}\right. $$

Step-by-Step Solution

Answer

The lines intersect at \( \left( \frac{7}{3}, \frac{7}{3}, \frac{2}{3} \right) \).

Step 1: Determine the direction vectors

Identify the direction vector of each line by looking at the coefficients of parameter \( t \). For line \( \ell_1 \), the direction vector is \( \mathbf{d_1} = \langle 2, -2, 1 \rangle \). For line \( \ell_2 \), the direction vector is \( \mathbf{d_2} = \langle -1, 5, 7 \rangle \).

Step 2: Check if lines are parallel

Two lines are parallel if their direction vectors are scalar multiples of each other. Check if \( \mathbf{d_2} = k \mathbf{d_1} \) for some scalar \( k \). The vectors \( \mathbf{d_1} = \langle 2, -2, 1 \rangle \) and \( \mathbf{d_2} = \langle -1, 5, 7 \rangle \) are not scalar multiples, so the lines are not parallel.

Step 3: Check for the same line or intersection

Substitute the parameter \( t \) from \( \ell_1 \) into the equations of \( \ell_2 \) to check if any parameter \( t \) satisfies both equations, indicating intersection. Set equations equal: \( 1+2t = 3-t \), \( 3-2t = 3+5t \), and \( t = 2+7t \). Solving these, we find \( t = \frac{2}{3} \) meets all conditions.

Step 4: Find the intersection point

Using \( t = \frac{2}{3} \), substitute into \( \ell_1 \): \( x = 1+2(\frac{2}{3}) = \frac{7}{3} \), \( y = 3-2(\frac{2}{3}) = \frac{7}{3} \), \( z=\frac{2}{3} \). The point of intersection is \( \left( \frac{7}{3}, \frac{7}{3}, \frac{2}{3} \right) \).

Step 5: Determine if lines are skew

Since the lines intersect, they are not skew. Skew lines would neither intersect nor be parallel, which does not apply here.

Key Concepts

Direction Vectors

Direction vectors are essential in vector calculus when dealing with lines in three-dimensional space. They describe the direction in which the line progresses. For a line defined parametrically, such as \( x = a + bt \), \( y = c + dt \), and \( z = e + ft \), the direction vector is \( \langle b, d, f \rangle \).

In the exercise, we have the lines \( \ell_1 \) with the direction vector \( \mathbf{d_1} = \langle 2, -2, 1 \rangle \) and \( \ell_2 \) with the direction vector \( \mathbf{d_2} = \langle -1, 5, 7 \rangle \). These vectors are derived from the coefficients of the parameter \( t \) from each line's parametric equations. They signify how the line moves in the 3D space as the parameter \( t \) varies.

Line Intersection

When two lines intersect, they meet at a single point in space. To check if two lines intersect, we solve the parametric equations of each line simultaneously. The goal is to find a common parameter value that satisfies both sets of equations.

For lines \( \ell_1 \) and \( \ell_2 \), this involves equating the parametric forms and solving them:

• Set \( 1 + 2t = 3 - t \)
• Set \( 3 - 2t = 3 + 5t \)
• Set \( t = 2 + 7t \)

After solving these, we found \( t = \frac{2}{3} \) satisfies all lines, implying that they intersect at a definite point. This shared point is calculated by substituting \( t = \frac{2}{3} \) into either line's parametric equations.

Skew Lines

Skew lines are lines that do not intersect and are not parallel. They exist in three-dimensional space but do not meet, essentially passing by each other at a certain distance. To verify whether lines are skew, check for both parallelism and intersection.

In our exercise, since the lines \( \ell_1 \) and \( \ell_2 \) were found to intersect, they are not skew. Skewness can only occur if lines do not fit the criteria of intersecting or being parallel. Understanding skew lines often involves visualizing lines in space, recognizing that despite their paths, they never actually cross each other.

Parallelism in Vector Calculus

Two lines are parallel in three-dimensional space if their direction vectors are scalar multiples of each other. This means that the vector describing the direction of one line can be obtained by multiplying the direction vector of the other line with a scalar \( k \).

To determine if lines \( \ell_1 \) and \( \ell_2 \) are parallel, we analyzed their direction vectors \( \mathbf{d_1} = \langle 2, -2, 1 \rangle \) and \( \mathbf{d_2} = \langle -1, 5, 7 \rangle \).

• If \( \mathbf{d_2} = k \mathbf{d_1} \), the lines are parallel.
• Here, no such \( k \) exists, proving the lines are not parallel.

Understanding parallelism helps in defining the geometry and arrangement of lines in space, ensuring a correct analysis of their relational properties in vector calculus.