Category: 11 Mathematics HL

Try to complete the following questions for Monday’s lesson.

1. Find the distance of the point  \(A(-1,1,2)\) to the line \[\displaystyle{\vec{r}=\begin{bmatrix}1\\-2\\3\end{bmatrix}+\lambda \begin{bmatrix}-2\\1\\1\end{bmatrix}}\]
2. The lines below intersect at a point \(A\). Use an algebraic method to find the coordinates of  \(A\), then verify your answer using GeoGebra.
   \[L_1: \vec{r}=\begin{bmatrix}-1\\-1\\4\end{bmatrix}+\lambda \begin{bmatrix}-2\\1\\1\end{bmatrix}\] \[L_2: \vec{r}=\begin{bmatrix}-2\\0\\7\end{bmatrix}+\lambda \begin{bmatrix}-6\\4\\8\end{bmatrix}\]
3. In 2D, any two non-parallel lines will have a point of intersection. In 3D, two lines can be non-parallel and have no point of intersection; such lines are called skew lines.
   Show that \(L_1\) from question 2 and \(L_3\) below are skew lines.
   \[L_3: \vec{r}=\begin{bmatrix}-2\\5\\12\end{bmatrix}+\lambda \begin{bmatrix}-6\\4\\8\end{bmatrix}\]

Find the distance of the point \(A(-3,1)\) to the line
\[\displaystyle{\vec{r}=\begin{bmatrix}1\\2\end{bmatrix}+\lambda \begin{bmatrix}-2\\3\end{bmatrix}}\]

using the method outlined below.

1. Find an expression for an arbitrary point \(D\) on the given line.
2. Using the expression you’ve produced, find a general expression for the vector \(\overrightarrow{AD}\).
3. Where \(\vec{b}\) is the direction vector of the given line, use the dot product \(\overrightarrow{AD}\cdot\vec{b}\) to find the coordinates of the point on the line closest to \(A\). Hence, find the distance from \(A\) to the line.

Complete the following question for our lesson tomorrow. In the first instance, use the Cartesian equation to find your solution, then see of you can find a vector-based solution to the same problem.

Find the distance of the point \(A(1,3)\) to the line
\[\displaystyle{\vec{r}=\begin{bmatrix}4\\2\end{bmatrix}+\lambda \begin{bmatrix}4\\1\end{bmatrix}}\]

Complete the following questions for tomorrow’s lesson.

1. Find a vector equation of the line passing through \(A(2,5)\) and \(B(6,-1)\).
2. Find a different vector equation for the same line.

As discussed in class last week, complete page 418 question 18 for tomorrow’s lesson.

For other examples of applications in physics, I suggest you also look at questions 28 and 29 on page 419.

Here’s a short question to consider tonight.

Given points A and B, let C be the midpoint of the line segment from A to B. Find an expression for \(\vec{c}\) in terms of  \(\vec{a}\) and  \(\vec{b}\) .

Complete pages 390–393 questions 12, 15, 17, 18, and 20 for tomorrow’s lesson.

Complete the questions on this document before the beginning of class on Monday, April 11th.

For this assignment, you should submit your work assignment electronically as a PDF file.