# Midterm 1 review

## General info

• Make sure to read all the information pertaining to the midterm in the News section of the forum on OWL
• Make sure to go to the right location. The location are divided by section number -- ours is 004 -- and first letter of last name
• Remember that this is a closed book exam. You can't bring a calculator, note sheets, textbooks, or any other aid.
• Make sure to bring your student ID, and an HB pencil to fill out the scantron with

## Writing the midterm

• Make sure that you wrote your name on each page of the exam.
• Make sure that you write your name, student number, etc on the scantron
• Start filling out the scantron in time! I'd recommend leaving at least 10 minutes for this.
• When you write the midterm, don't spend too much time on one problem. You should definitely not spend more than 5 minutes on one problem. In fact, if you don't immediately know how to solve a problem, I'd recommend skipping it, doing what you can immediately do, and only returning if you have time.
• There are some practice exams on OWL. I'd highly recommend to practice timing yourself on these. Set an alarm for 90 minutes, make sure that you're not using any aid, simulate the exam experience!

## Review of midterm material

A point $$P$$ in Euclidean $$n$$-space $$\mathbf R^n$$ is an $$n$$-tuple of numbers $$P(p_1,\dotsc,p_n)$$. The numbers $$p_1,\dotsc,p_n$$ are called the coordinates of $$P$$. We call $$\mathbf R^2$$ the plane, and $$\mathbf R^3$$ the (3-)space.

A vector in $$\mathbf R^n$$, or an $$n$$-vector is also described by an $$n$$-tuple $$\mathbf u=(u_1,\dotsc,u_n)$$, but in this case the numbers $$u_1,\dotsc,u_n$$ are called the components of $$u$$.

A vector can be represented by an arrow between two points: $$\vec{PQ}=(q_1-p_1,\dotsc,q_n-p_n)$$. We say that $$P$$ is the source, and $$Q$$ is the target. A vector $$\mathbf u$$ can always be represented as the arrow $$\vec{OP}$$, where $$O(0,\dotsc,0)$$ is the origin, and $$P(u_1,\dotsc,u_n)$$.

We have defined addition, scalar multiplication and difference componentwise: $\mathbf u+\mathbf v=(u_1+v_1,\dotsc,u_n+v_n),\quad c\mathbf u=(cu_1,\dotsc,cu_n),\quad\mathbf u-\mathbf v=(u_1-v_1,\dotsc,u_n-v_n).$

In $$\mathbf R^3$$, we have the standard 3-vectors $$\mathbf i=(1,0,0),\mathbf j=(0,1,0),\mathbf k=(0,0,1)$$. Using these, we can write for any 3-vector $$\mathbf u$$ $\mathbf u=u_1\mathbf i+u_2\mathbf j+u_3\mathbf k.$ By restriction to $$\mathbf R^2$$ as the plane $$z=0$$, we have the standard 2-vectors $$\mathbf i=(1,0),\mathbf j=(0,1)$$.

We define Euclidean distance as $d(P,Q)=\sqrt{(q_1-p_1)^2+\dotsb+(q_n-p_n)^2}.$

The length (magnitude, norm) of a vector $$\mathbf u$$ is $\|u\|=\sqrt{u_1^2+\dotsb+u_n^2}.$ Note that this is the length of any arrow representing $$\mathbf u$$.

The dot product of two vectors is $\mathbf u\cdot\mathbf v=u_1v_1+\dotsb+u_nv_n.$

Let $$\mathbf u,\mathbf v$$ be two nonzero vectors. Their angle $$\alpha$$ can be given by $\mathbf u\cdot\mathbf v=\cos\alpha\|\mathbf u\|\cdot\|\mathbf v\|.$ Accordingly, we say that $$\mathbf u$$ and $$\mathbf v$$ are orthogonal, if $$\mathbf u\cdot\mathbf v=0$$.

We say that $$\mathbf u$$ and $$\mathbf v$$ are parallel, if there exists a scalar $$c$$ such that $$c\mathbf u=\mathbf v$$. If $$c>0$$, then they have the same direction. If $$c<0$$, then they have the opposite direction.

A unit vector is a vector with length 1. Let $$\mathbf u$$ be a nonzero vector. Then the unit vector in the same direction is $$\frac{1}{\|\mathbf u\|}\mathbf u$$.

Let $$\mathbf u$$ and $$\mathbf v$$ be 3-vectors. Then their cross product is $\mathbf u\times\mathbf v=\begin{vmatrix} \mathbf i & u_1 & v_1 \\ \mathbf j & u_2 & v_2 \\ \mathbf k & u_3 & v_3 \end{vmatrix}=(u_2v_3-u_3v_2,u_3v_1-u_1v_3,u_1v_2-u_2v_1).$ The cross product $$\mathbf u\times\mathbf v$$ is orthogonal to both $$\mathbf u$$ and $$\mathbf v$$.

The area of the parallelogram selected by $$\mathbf u$$ and $$\mathbf v$$ is $$\|\mathbf u\times\mathbf v\|$$.

The triple product is $$\mathbf u(\mathbf v\times\mathbf w)$$. The volume of the parallelpiped selected by $$\mathbf u,\mathbf v,\mathbf w$$ is $$|\mathbf u(\mathbf v\times\mathbf w)|$$.

Let $$\ell$$ be a line. Then a vector $$\mathbf u$$ is a direction vector of $$\ell$$, if $$\mathbf u=\vec{PQ}$$ for two points $$P,Q$$ on $$\ell$$. Note that in this case $$\mathbf v$$ is also a direction vector of $$\ell$$ precisely when it is parallel to $$\mathbf u$$. That is, two lines are parallel precisely when they have the same direction vectors.

Let $$P$$ be a point on $$\ell$$, and $$\mathbf u$$ a direction vector of $$\ell$$. Then these give a point-parallel form equation for $$\ell$$: $\mathbf r(t)=\mathbf p+t\mathbf u.$

Let $$P$$ and $$Q$$ be two distinct points on $$\ell$$. Then these give a two-point form equation for $$\ell$$: $\mathbf r(t)=(1-t)\mathbf p+t\mathbf q.$

Let $$P$$ be a point and $$\mathbf n$$ a vector. Then the collection of points $$X$$ satisfying the point-normal form equation $\mathbf n\cdot(\mathbf x-\mathbf p)=0$ is a hyperplane $$\Pi$$. A hyperplane in $$\mathbf R^3$$ is a plane. A hyperplane in $$\mathbf R^2$$ is a line. The corresponding standard form equation is $n_1x_1+\dotsb+n_nx_n=n_1p_1+\dotsb n_np_n.$

That is, in $$\mathbf R^2$$, a line $$\ell$$ can be given either by

• a point $$P$$ and a direction vector $$\mathbf u$$,
• or a point $$P$$ and a normal vector $$\mathbf n$$.

Let $$\mathbf u$$ be a direction vector of $$\ell$$. Then $$\mathbf n$$ is a normal vector precisely when it is orthogonal to $$\mathbf u$$. An easy choice for $$\mathbf n$$ is $$(-u_2,u_1)$$.

Suppose that $$\mathbf u$$ is a direction vector of $$\ell$$, and $$\mathbf n$$ is a normal vector of $$\ell$$. Let $$\ell_2$$ be a line perpendicular to $$\ell$$. Then $$\mathbf u$$ is a normal vector of $$\ell_2$$, and $$\mathbf v$$ is a direction vector of $$\ell_2$$.

Let $$P,Q,R$$ be three noncollinear (not on the same line) points in $$\mathbf R^3$$. Let $$\Pi$$ be the plane containing $$P,Q,R$$. Then an easy choice for a normal vector of $$\Pi$$ is $$\vec{PQ}\times\vec{PR}$$.

Back to $$\mathbf R^n$$ with any $$n\ge2$$. Let $$\ell_1$$ have point $$P$$ and direction vector $$\mathbf u$$. Let $$\ell_2$$ have point $$Q$$ and direction vector $$\mathbf v$$. Then the intersections of $$\ell_1$$ and $$\ell_2$$ are of the form $\mathbf p+s\mathbf u\text{ or equally }\mathbf q+t\mathbf v,$ where the $$(s,t)$$ are the solutions of the vector equation $\mathbf p+s\mathbf u=\mathbf q+t\mathbf v.$