Question

Line I has equation $${\mathit{r}}_1=\left(\begin{array}{l}2\\ 3\\ 5\end{array}\right)+k\left(\begin{array}{l}1\\ 2\\ 4\end{array}\right)$$ Line II has equation $${\mathit{r}}_2=\left(\begin{array}{c}-5\\ 8\\ 1\end{array}\right)+l\left(\begin{array}{c}-6\\ 7\\ 0\end{array}\right)$$ Different values of $k$ give different points on line I. Similarly, different values of $l$ give different points on line II. If the two lines intersect then ${\mathit{r}}_1={\mathit{r}}_2$ at the point of intersection. If you can find values of $k$ and $l$ which satisfy this condition then the two lines intersect. Show the lines intersect by finding these values and hence find the point of intersection.

Short answer

Line I: $${\mathit{r}}_1=\left(\begin{array}{l}2\\ 3\\ 5\end{array}\right)+k\left(\begin{array}{l}1\\ 2\\ 4\end{array}\right)$$ Line II: $${\mathit{r}}_2=\left(\begin{array}{c}-5\\ 8\\ 1\end{array}\right)+l\left(\begin{array}{c}-6\\ 7\\ 0\end{array}\right)$$ Answer: Yes, the given lines intersect at the point P$\left(\frac{3}{2}, 2, \frac{1}{2}\right)$.

Step-by-step solution

Compare the Parametric Equations

First, let's analyze the given equations for the two lines: Line I: $${\mathit{r}}_1=\left(\begin{array}{l}2\\ 3\\ 5\end{array}\right)+k\left(\begin{array}{l}1\\ 2\\ 4\end{array}\right)$$ Line II: $${\mathit{r}}_2=\left(\begin{array}{c}-5\\ 8\\ 1\end{array}\right)+l\left(\begin{array}{c}-6\\ 7\\ 0\end{array}\right)$$ For the two lines to intersect, they must satisfy the condition ${\mathit{r}}_1={\mathit{r}}_2$ at the point of intersection.

Set Equations Equal and Solve

Equating the two lines, we get: $$\left(\begin{array}{l}2\\ 3\\ 5\end{array}\right)+k\left(\begin{array}{l}1\\ 2\\ 4\end{array}\right)=\left(\begin{array}{c}-5\\ 8\\ 1\end{array}\right)+l\left(\begin{array}{c}-6\\ 7\\ 0\end{array}\right)$$ Now, we can write the equation components-wise: $$2+k=-5-6l$$ $$3+2k=8+7l$$ $$5+4k=1$$

Solve the System of Equations

Now we'll solve for $k$ using the third equation: $$5+4k=1⟹k=\frac{-1}{2}$$ Now using the value of $k$, we'll solve for $l$, substituting $k$ into the first equation: $$2+\frac{-1}{2}=-5-6l⟹l=\frac{1}{6}$$

Find the Point of Intersection

Since we found values for $k$ and $l$, the two lines intersect. To find the point of intersection, we can substitute the value of $k$ into the equation for Line I: $${\mathit{r}}_1=\left(\begin{array}{l}2\\ 3\\ 5\end{array}\right)+\frac{-1}{2}\left(\begin{array}{l}1\\ 2\\ 4\end{array}\right)=\left(\begin{array}{l}\frac{3}{2}\\ 2\\ \frac{1}{2}\end{array}\right)$$ The point of intersection is: $$P(\frac{3}{2},2,\frac{1}{2})$$

Key concepts

Parametric Equations

Understanding parametric equations is like learning the language of lines and curves in mathematics. These equations relate two or more sets of variables to one or more independent parameters. For lines in three-dimensional space, like Line I and Line II in the exercise, parametric equations provide an elegant way to describe not just where a line is, but how it extends infinitely in both directions.

In our example, the values of k and l act as the parameters for Lines I and II, respectively. The components of the vectors are the coordinates of the points on the line. As k and l vary, they trace out the line, giving us a way to find any point along it. Parametric equations are essential because they give us the power to explore lines in space without restricting us to a single viewpoint or dimension.

System of Equations

A system of equations consists of multiple equations that share variables and are solved simultaneously. The goal is to find the common solution that satisfies all equations in the system. Picture it like a treasure hunt, where each clue (equation) gets you closer to the treasure (solution).

In the quest to discover the point of intersection for two lines, setting their parametric equations equal to each other creates a system of equations. Each of the individual component equations involves k and l and represents a balance that needs to be achieved across the x, y, and z coordinates for the lines to intersect. But remember, finding a solution doesn't always guarantee intersection; this only happens if the solution yields reasonable values of the parameters, within the context of our lines in space.

Point of Intersection

The point of intersection is the GPS pin on the map, marking the exact spot where the paths of Line I and Line II cross. This point is significant not just because it is a shared point, but also because it signifies a unique set of parameter values, k and l, which validate the concurrent location.

Finding the point of intersection is like solving a mystery — once the values of k and l have been deduced from the system of equations, plugging them back into any line's parametric equation will reveal the coordinates of the intersection. The exercise leads us methodically to this discovery, ensuring students not only get the 'right' answer but also understand the process which is critical for handling more complex scenarios in advanced mathematics.