Midterm 1 review

General info

Writing the midterm

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

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 \[ \mathbf p+s\mathbf u=\mathbf q+t\mathbf v. \]