2.1 Vectors in \(\mathbf R^m\)

As it hath been foretold, now we'll generalize the theory of vectors in 2- and 3-dimensional space to \(m\)-dimensional space for any positive integer \(m\).

Definition. Euclidean \(m\)-space \(\mathbf R^m\) is the collection of \(m\)-dimensional points, that is \(m\)-tuples of numbers \(P(x_1,\dotsc,x_m)\). The numbers \(x_i\) are called the coordinates of the point. A vector in \(\mathbf R^m\) (\(m\)-vector) can also be described as an \(m\)-tuple \(\mathbf v=(v_1,\dotsc,v_m)\). For a vector, the numbers \(x_i\) are called its components. The zero vector in \(\mathbf R^m\) is \(\boldsymbol0=(0,\dotsc,0)\). Two vectors \(\mathbf v,\mathbf w\) are equal, if all their components are equal: \(v_i=w_i,\,1\le i\le m\).

Remarks. (1) For typographical reasons I had to switch to horizontal notation when writing vectors by components inline. I still find column vectors more intuitive.

  1. In \(\mathbf R^m\) too vectors can be defined as arrows where two are equal when they are translates.

  2. With \(m\)-dimensional vectors too we adopt the notation that if I say that \(\mathbf v\) is a vector in \(\mathbf R^m\), then that implies that \(v_1,\dotsc,v_m\) denote its components.

Definition. Let \(P(p_1,\dotsc,p_m),Q(q_1,\dotsc,q_m)\) be two points in \(\mathbf R^m\). Then their distance is \[ d(P,Q)=\sqrt{(p_1-q_1)^2+\dotsb+(p_m-q_m)^2}. \] Let \(\mathbf v=(v_1,\dotsc,v_m)\) be a vector in \(\mathbf R^m\). Then its length (magnitude, norm) is \[ \|\mathbf v\|=\sqrt{v_1^2+\dotsb+v_m^2}. \] A vector in \(\mathbf R^m\) is a unit vector, if it has length 1.

Remark. We say that \(\mathbf R^m\) is Euclidean \(m\)-space, because it is equipped with the above distance function. As another example (which of course is not covered in this class), special relativity theory is formulated in Minkowski space. That is \(\mathbf R^4\), but the distance function is different: \[ d(P(x_1,y_1,z_1,t_1),Q(x_2,y_2,z_2,t_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2-(t_1-t_2)^2}. \] Note that unlike Euclidean space, it is possible to get negative distances in Minkowski space. Vectors with negative length are called timelike, and vectors with positive length are called spacelike.

Exercise. Out of the following vectors, which is a unit vector? \[ \left(\begin{smallmatrix}2\\0\\-1\\-1\end{smallmatrix}\right),\,\left(\begin{smallmatrix}0\\1/3\\0\\1/3\\0\\1/3\end{smallmatrix}\right),\,\left(\begin{smallmatrix}2/3\\0\\2/3\\0\\1/3\end{smallmatrix}\right),\,\left(\begin{smallmatrix}-1/2\\0\\1/2\end{smallmatrix}\right). \]

Definition. Let \(c\) be a scalar, and \(v\) an \(m\)-vector. Then the scalar multiple is \(c\mathbf v=(cv_1,\dotsc,cv_m)\). Two \(m\)-vectors \(\mathbf u,\mathbf v\) are parallel (collinear) if one is a scalar multiple of the other. Suppose that the \(\mathbf u,\mathbf v\) are parallel nonzero vectors. Then there exists a scalar \(c\) such that \(c\mathbf u=\mathbf v\). If \(c>0\) then we say \(\mathbf u\) and \(\mathbf v\) have the same direction, and if \(c<0\) then we say they have the opposite direction.

Theorem. Let \(c\) be a scalar, and \(v\) an \(m\)-vector. Then we have \[ \|c\mathbf v\|=|c|\cdot\|\mathbf v\|. \]

Exercise. Find the unit vector with the opposite direction of \((9,-3,0,12)\).

Definition. Let \(\mathbf u,\mathbf v\) be \(m\)-vectors. Then their vector sum is \(\mathbf u+\mathbf v=(u_1+v_1,\dotsb,u_m+v_m)\). Their vector difference is \(\mathbf u-\mathbf v=(u_1-v_1,\dotsc,u_m-v_m)\). Their dot product is \(\mathbf u\cdot\mathbf v=u_1v_1+\dotsb+u_mv_m\). We say that \(\mathbf u\) and \(\mathbf v\) are orthogonal, if we have \(\mathbf u\cdot\mathbf v=0\).

Remark. As we have said before, if \(m\ne3\), then there is no cross product for \(m\)-vectors.

Exercise Find \(k\) so that the vectors \(2(1,3,0,2)-(0,-1,2,0)\) and \((a,2,0,1)\) are orthogonal.

Definition. Let \(P,Q\) be two different points in \(\mathbf R^m\). Let \(\ell\) be the line through \(P\) and \(Q\). Let \(\mathbf p=\vec{OP},\,\mathbf q=\vec{OQ}\), and \(\mathbf v=\vec{PQ}\). A two-point form equation of \(\ell\) is \[ \mathbf r(t)=t\mathbf q+(1-t)\mathbf p. \] A point-parallel form equation of \(\ell\) is \[ \mathbf r(t)=\mathbf p+t\mathbf v. \] Writing out the component functions of a parametrization, we get parametric equations for \(\ell\): \[\begin{align*} r_1(t)=&p_1+tv_1\\ \vdots&\\ r_m(t)=&p_m+tv_m. \end{align*}\]

Exercise. Let \(\ell\) be the line through \(P(0,3,-1,2)\) and \(Q(1,-3,0,1)\). Write down a two-point form equation, a point-parallel form equation, and parametric equations for it.

Definition. Let \(P\) be a point in \(\mathbf R^m\), and \(\mathbf n\) an \(m\)-vector. Then the hyperplane through \(P\) with normal vector \(\mathbf n\) is the collection of points \(Q(x_1,\dotsc,x_m)\) the coordinates of which satisfy the point-normal equation \[ \mathbf n\cdot(\mathbf x-\mathbf p)=0, \] or the standard form equation \[ n_1x_1+\dotsb+n_mx_m=c, \] where \(c=\mathbf n\cdot\mathbf p\).

Exercise. Find a normal vector for the hyperplane with standard form equation \[ 3x_1+x_3-2x_5=15. \] Find a point on this hyperplane.