11 HL: Planes and Normals

Complete the questions in the PDF file attached before next class. (Questions 1 b) and 3 a) are challenge questions, but you might be able to figure these out using the material we’ve covered in the past few lessons… see the hint below.)

A line with just the right direction, in just the right location, might intersect the plane in just the right point…

11 HL: Vector Equations of Lines in 3D

Complete the following questions before our next class.

Exercise 13A questions 4, 5ac, 6ab

Also, (as a challenge question) can you answer question 2 from class? Both questions are shown below.

  1. Find the vector equation of the line passing through \(A(1,-1,4)\) and \(B(3,3,-1)\).
  2. Find the distance of the point \(A(1,2,3)\) to the line given by
    \[\vec{r}=\begin{bmatrix}4\\2\\1\end{bmatrix}+\lambda \begin{bmatrix}4\\1\\-1\end{bmatrix}\]

12 SL: Antiderivatives

As I’m away on Monday, you’ll want to get a head start on antiderivatives, which are, in some sense, the opposite of derivatives. For example, if the derivative of \(x^2\) is \(2x\), then the antiderivative of \(2x\) is \(x^2+c\), where \(c\) is the constant of integration.

Read Section 18B, 18D, and 18E.1 (paying particular attention to the examples), and complete the following questions.

Exercise 18B questions 1–3
Exercise 18D questions 1–9, 11, 12
Exercise 18E.1 questions 1–3

Aim to complete most, if not all, of these questions before our next class.

9 Extended: General Form Equation of a Line

If we are given a point on a line, and the slope of that line, we already know how to find the equation of that line in slope-intercept form. How can we find the general form equation of that same line?

In one sense it’s very easy, we can just move the x-term to the same side as the y-term. For example, the line with slope-intercept form \[\begin{align}y&=3x+1 \quad\textrm{can also be written as}\\y-3x&=1\end{align}\]

which is in general form.

There is, however, another easy way to find either the general form equation of a line that you may also want to use. If you want to learn that method, have a look at page 174 in our textbook. You can use any method you like!

Complete the following questions before our next class.

Exercise 9E.1 questions 1–3, 4ef, 5, and 7 (if you want a challenge, try 11 as well!).