Question

Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection. a. \(x=1+s, y=2 s\) and \(x=1+2 t, y=3 t\) b. \(x=2+5 s, y=1+s\) and \(x=4+10 t, y=3+2 t\) c. \(x=1+3 s, y=4+2 s\) and \(x=4-3 t, y=6+4 t\)

Short answer

Question: Determine whether the given pairs of lines are parallel, intersecting, and if intersecting, find the point of intersection. a. Line 1: (1+s, 2s), Line 2: (1+2t, 3t) b. Line 1: (5s, 1+s), Line 2: (10t, 2t) c. Line 1: (1+3s, 4+2s), Line 2: (4-3t, 6+4t) Answer: a. Intersecting at point (1,0). b. Parallel. c. Intersecting at point (4,6).

Step-by-step solution

For each pair of lines, identify the direction vectors by looking at the coefficients of the parameters 's' and 't'. a. Direction vector for line 1: \((1,2)\), line 2: \((2,3)\). b. Direction vector for line 1: \((5,1)\), line 2: \((10,2)\). c. Direction vector for line 1: \((3,2)\), line 2: \((-3,4)\). #Step 2: Check if the lines are parallel#

Check if the direction vectors for each pair of lines are proportional. If they are, then the lines are parallel. a. The direction vectors for lines 1 and 2 are \((1,2)\) and \((2,3)\), which are not proportional. Therefore, the lines are not parallel. b. The direction vectors for lines 1 and 2 are \((5,1)\) and \((10,2)\), which are proportional. Therefore, the lines are parallel. c. The direction vectors for lines 1 and 2 are \((3,2)\) and \((-3,4)\), which are not proportional. Therefore, the lines are not parallel. #Step 3: Find the point of intersection for the non-parallel lines#

For the non-parallel lines (pairs a and c), find the value of the parameters 's' and 't' that make the parametric equations equal. Pair a: Equate the x components: \(1+s = 1+2t\) \(s = 2t\) Equate the y components: \(2s = 3t\) Substitute the value of s from the previous equation: \(2(2t) = 3t\) \(4t = 3t\) \(t = 0\) \(s = 2t = 2(0) = 0\) Intersection point: \((1 + 0, 2(0)) = (1,0)\) Pair c: Equate the x components: \(1+3s = 4-3t\) \(3s + 3t = 3\) \(s + t = 1\) Equate the y components: \(4+2s = 6+4t\) \(2s - 4t = 2\) \(s - 2t = 1\) From the first equation, \(s = 1 - t\), substitute into the second equation: \(1 - t - 2t = 1\) \(-3t = 0\) \(t = 0\) \(s = 1 - t = 1 - 0 = 1\) Intersection point: \((1 + 3(1), 4 + 2(1)) = (4,6)\) #Step 4: Present the results#

a. The lines are intersecting at the point \((1,0)\). b. The lines are parallel. c. The lines are intersecting at the point \((4,6)\).

Key concepts

Direction Vectors

Understanding direction vectors is crucial when working with lines in geometry. They provide us insight into the lines' direction. Simply put, the direction vector of a line can be seen as a way to measure which way and how fast we move from one point on the line to another. Generally expressed as ordered pairs or triples in space, these vectors are derived from the coefficients of the parameters in the parametric equations.
For example, in the equation \( x = 1 + s \) and \( y = 2s \), the direction vector is \((1,2)\). Here, we can think of '1' as the change in the x-direction and '2' as the change in the y-direction with respect to 's'.
The role of direction vectors is especially pivotal in determining whether two lines are parallel. If the direction vectors of two lines are proportional, meaning one can be an exact multiple of the other, the lines are indeed parallel. For instance, with direction vectors \((5,1)\) and \((10,2)\), since \(10/5 = 2/1\), the vectors are proportional, indicating parallel lines.

• Key Point: Direction vectors tell how lines move in space.
• They determine if lines are parallel or not.

Parametric Equations

Parametric equations provide a formulaic way of representing lines using parameters, often denoted by letters like 's' or 't'. These parameters essentially act as the 'dials' which, when adjusted, trace out the location of points on the line.
Each line can be expressed using parametric equations, such as \(x = 1 + s\) and \(y = 2s\). Here, the parameter 's' modifies the x and y, starting from a point like (1, 0) and moving along the direction vector \((1, 2)\). This approach allows for a flexible way to represent lines, from simple straight lines to more complex paths in 3D space.

• Parametric equations define the path of a line through its parameters.
• They provide a more dynamic representation compared to traditional line equations.

The beauty of parametric equations lies in their ability to accommodate changes in representation, making them crucial in determining intersections between lines.

Point of Intersection

Finding the point of intersection of two lines involves setting their parametric equations equal to each other. This process identifies the exact coordinates at which two non-parallel lines meet. Let's break it down with an example:
For intersecting lines with equations \(x = 1+s\), \(y = 2s\) and \(x = 1+2t\), \(y = 3t\), we start by equating the respective components:

• Equate x-components: \(1+s = 1+2t\). Solving this yields \(s = 2t\).
• Equate y-components: \(2s = 3t\). Substitute \(s = 2t\) into this to simplify: \(2(2t) = 3t\), gives \(t = 0\) and \(s = 0\) as they equate to zero.

Ultimately, substituting back these 's' and 't' into the x and y equations leads us to the intersection point: \( (1+0, 2(0)) = (1, 0)\).

This process helps us find the precise point where two lines can be expected to meet, provided they aren't running parallel to each other. Always equate the same components and solve systematically for the cleanest route to the solution, helping you unlock exactly where and how two paths cross.