6.7 Vectors and Coordinate Systems: Glossary

Costas Loizou; Shuwei Chen; Deak Kirkham

6.7 Vectors and Coordinate Systems: Glossary

The pronunciation guide given in these glossaries draws on the symbols International Phonetic Alphabet. We have made two adaptations to the IPA system. Firstly, we use slanted lines as opposed to square brackets to enclose pronunciation material. We do this as the slanted line convention is more common in language teaching contexts. Secondly, we use the symbol <r> to denote the voiced alveolar approximant sound which appears in British English at the beginning of the each word in the phrase 'red rhombus'; the IPA would instead use an inverted 'r' symbols namely <ɹ>. We have done this to avoid unnecessary complication to the visuals of the glossaries.

In addition to these conventions, please also note that the pronunciations given are indicative of contemporary Standard British English pronunciation and should not be read as normative, nor as a target pronunciation that you must aim to produce. The pronunciation of English varies widely across World Englishes; part of becoming fluent in the academic English of mathematics is developing an awareness of, and sensitivity to, the variation in pronunciation that you will encounter.

Word	Pronunciation	Part of speech词性	Related terminology	Chinese meaning	Example sentence 1	Example sentence 2
vector	/ˈvektə(r)/	n.	unit vector 单位向量； basis vector 基向量； vector field 向量场	向量，矢量	We can add two vectors using the rule \({\bf a}+ {\bf b}=(a_1+b_1, a_2+b_2)\). 我们可以用法则\({\bf a}+ {\bf b}=(a_1+b_1, a_2+b_2)\)来相加两个向量。	A vector has both magnitude and direction. 向量具有大小和方向。
scalar	/ˈskeɪlə(r)/	n.	scalar field 标量场	标量，数量	In the expression \(5 {\bf a}\), the number \(5\) acts as a scalar multiplier. 在表达式 \(5 {\bf a}\) 中数字 \(5\) 充当标量乘子。	Multiplying a vector \({\bf v}\) by a scalar \(\lambda\) gives \(\lambda {\bf v}\). 用标量 \(\lambda\)乘以向量\({\bf v}\) 得到\(\lambda {\bf v}\)。
direction vector	/daɪˈrɛkʃən ˈvɛktə/	n. phr.		方向向量	The line passes through \(P_0\) with direction vector \({\bf d} = (2,-1,3)\). 该直线过点 \(P_0\) 且方向向量为 \({\bf d} = (2,-1,3)\)。	Two parallel lines share the same direction vector. 两条平行线具有相同的方向向量。
normal vector	/ˈnɔːməl ˈvɛktə/	n. phr.		法向量	The plane \(ax + by + cz = d\) has normal vector \({\bf n} = (a,b,c)\). 平面 \(ax + by + cz = d\) 的法向量为 \({\bf n} = (a,b,c)\)。	Use cross product \({\bf u} \times {\bf v}\) to find a normal vector. 利用叉积 \({\bf u} \times {\bf v}\) 求解一个法向量。
in-plane vector	/ɪn pleɪn ˈvɛktə/	n. phr.		平面内向量	An in-plane vector is orthogonal to the plane’s normal vector \({\bf n}\). 平面内向量与该平面的法向量 \({\bf n}\) 正交。	The tangent vector of a planar curve is an in-plane vector. 平面曲线的切向量属于平面内向量。
component	/kəmˈpəʊnənt/	n.		分量，组成部分	The \(z\)-component of the vector \((2,-1,5)\) is \(5\). 向量\((2,-1,5)\)的\(z\)分量是\(5\)。	The component of \({\bf v}=(3,7)\) along the \(x\)-axis is \(3\). 向量\({\bf v}=(3,7)\)在\(x\)轴方向的分量是\(3\)。
coordinate	/kəʊˈɔːdɪnət/	n.	Coordinate system 坐标系； Coordinate Transformation 坐标转换； Cordinate-free method 无坐标方法	坐标	The point with coordinates \((-2,3)\) lies in the second quadrant. 坐标为\((-2,3)\)的点位于第二象限。	Each point on a plane is identified by a pair of coordinates \((x,y)\). 平面上的每个点都由一对坐标\((x,y)\)确定。
Cartesian	/kɑːˈtiːziən/	adj.	Descartes 笛卡尔； Cartesian coordinates 笛卡尔坐标系（直角坐标系）	笛卡尔的	Vectors are easily expressed in Cartesian components. 向量可以方便地用笛卡尔分量表示。	Cartesian coordinates use a grid of perpendicular \(x\) and \(y\) axes. 笛卡尔坐标使用相互垂直的\(x\)轴和\(y\)轴构成的网格。
rectangular coordinates	/rekˈtæŋɡjələ(r) kəʊˈɔːdɪnəts/	n. phr.	cylindrical coordinates 柱坐标系； spherical coordinates 球坐标系	直角坐标	We can convert polar form to rectangular coordinates using \(x=r \cos \theta\). 我们可以用\(x=r \cos \theta\)将极坐标转换为直角坐标。	In rectangular coordinates the point \((3,4)\) is \(3\) units right and \(4\) up. 在直角坐标系中点\((3,4)\)表示向右\(3\)个单位向上\(4\)个单位。
axis (plural: axes)	/ˈæksɪs/, /ˈæksiːs/	n.	axis of rotation 旋转轴； horizontal/vertical axis 水平/垂直轴； real axis 实轴； imaginary axis 虚轴	坐标轴	The axis of symmetry divides the graph. 对称轴将图像分成两部分。	The \(x\)-axis is horizontal in the coordinate plane. \(x\)轴在坐标平面中是水平的。
origin	/ˈɒrɪdʒɪn/	n.		原点	Distance is measured from the origin. 距离从原点开始测量。	The origin is the point \((0,0)\) in the coordinate system. 原点是坐标系中的\((0,0)\)。
quadrant	/ˈkwɒdrənt/	n.		象限	The point \((3,4)\) lies in the first quadrant. 点 \((3,4)\) 位于第一象限。	Solve \(\sin \theta \gt 0\), \(\cos \theta \lt 0\) in the second quadrant. 在第二象限求解 \(\sin \theta \gt 0\), \(\cos \theta \lt 0\)。
Euclidean space	/juːˈklɪdiən speɪs/	n. phr.	Euclidean geometry 欧几里得几何； non-Euclidean geometries 非欧几何	欧几里得空间	Parallel lines never meet in Euclidean space. 在欧几里得空间中平行线永不相交。	Three-dimensional Euclidean space is denoted \(\mathbb{R}^3\). 三维欧几里得空间记作\(\mathbb{R}^3\)。
dot product	/dɒt ˌprɒdʌkt/	n. phr.	scalar product 数量积	点积	We use the dot product to find the angle between two vectors. 我们用点积来求两个向量之间的夹角。	The dot product is calculated as \({\bf a} \cdot {\bf b}=a_1 b_1+a_2 b_2\). 点积计算为\({\bf a} \cdot {\bf b}=a_1 b_1+a_2 b_2\)。
cross product	/krɒs ˌprɒdʌkt/	n. phr.	vector product 向量积	叉乘	We use the cross product to find the area of a parallelogram. 我们用叉积来求平行四边形的面积。	The cross product of two vectors produces a new vector. 两个向量的叉积得到一个新的向量。
scalar triple product	/ˈskeɪlə ˈtrɪpl ˌprɒdʌkt/	n. phr.	vector triple product 向量三重积	标量三重积	The value of the scalar triple product gives the volume of a parallelepiped. 标量三重积的值表示平行六面体的体积。	The scalar triple product is \({\bf a} \cdot ({\bf b} \times {\bf c})\). 标量三重积为\({\bf a} \cdot ({\bf b} \times {\bf c})\)。
line	/laɪn/	n.		直线，线	The line passes through the origin. 这条直线经过原点。	Two points determine a line. 两点确定一条直线。
plane	/pleɪn/	n.		平面	The triangle lies on a plane. 这个三角形位于一个平面上。	Two lines in the same plane may intersect. 同一平面内的两条直线可能相交。
parallel	/ˈpærəlel/	adj.		平行的	These two lines are parallel and never meet. 这两条直线平行且永不相交。	Parallel lines remain the same distance apart. 平行线之间的距离始终相等。
perpendicular	/ˌpɜːpənˈdɪkjələ(r)/	adj.		垂直的	The two lines are perpendicular to each other. 这两条直线互相垂直。	The wall is perpendicular to the floor. 墙面与地面垂直。
orthogonal	/ɔːˈθɒɡənl/	adj.		正交的	Orthogonal lines meet at right angles. 正交直线相交成直角。	Two vectors are orthogonal if their dot product is zero. 两个向量点积为\(0\)则正交。
translation	/trænzˈleɪʃən/	n.		平移	The translation map is \(T(x)=x+c\). 平移变换表达式为 \(T(x)=x+c\)。	Translation shifts all points by a constant vector \({\bf c}\). 平移将所有点按常向量 \({\bf c}\) 偏移。
reflection	/rɪˈflɛkʃən/	n.		反射	Reflection preserves the length of any vector \(\\| {\bf u} \\|\). 反射保持任意向量的模长 \(\\| {\bf u} \\|\) 不变。	Composing two reflections yields a rotation. 两次反射复合可得到旋转变换。
rotate	/rəʊˈteɪt/	v.	rotation n. 旋转； revolution 旋转（绕轴转一圈）	旋转	The point rotates around the origin. 点绕原点旋转。	Rotate the shape by \(90\) degrees. 将图形旋转\(90\)度。
zoom	/zuːm/	v.		缩放	Zoom the coordinate system by scaling \(x \rightarrow 2x\), \(y \rightarrow 2y\). 通过缩放变换 \(x \rightarrow 2x\), \(y \rightarrow 2y\) 对坐标系进行放大。	A similarity transformation can zoom the figure uniformly. 相似变换可对图形进行均匀缩放。
stretched	/stretʃt/	adj.		拉伸的，延展的	A vector is stretched when multiplied by a scalar with magnitude greater than 1. 向量在与模大于1的标量相乘时会被拉伸。	If \(\lambda \gt 1\) the vector \(\lambda {\bf v}\) is stretched along its original direction. 如果 \(\lambda \gt 1\)向量\(\lambda {\bf v}\)会沿其原方向被拉伸。
projection	/prəˈdʒɛkʃən/	n.	project v.；projective adj. 射影的，投影的	投影	The shadow is the vertical projection onto the \(xy\)-plane. 该阴影是向 \(xy\) 平面所作的垂直投影。	Use projection to decompose vector into two perpendicular components. 利用投影将向量分解为两个互相垂直的分量。
normalise	/ˈnɔːməlaɪz/	v.	normalisation n. 归一化（过程）	归一化，标准化	We often normalise vectors before computing projections. 在计算投影之前我们通常会对向量进行归一化。	We normalise \({\bf v}\) to get the unit vector \({\bf \hat{v}}=\frac {{\bf v}}{\|{\bf v}\|}\). 我们对\({\bf v}\)进行归一化得到单位向量\({\bf \hat{v}}=\frac {{\bf v}}{\|{\bf v}\|}\)。
factorise	/ˈfæktəraɪz/	v.	factorisation n. 因式分解	因式分解	To factorise \(x^2-4\) we write \((x-2)(x+2)\). 对\(x^2-4\)进行因式分解得到\((x-2)(x+2)\)。	Factorise the polynomial completely to find its roots. 将多项式完全因式分解以求其根。
invariant	/ɪnˈveərɪənt/	n.	geometric invariant 几何不变量	不变量	A geometric invariant does not change under coordinate changes. 几何不变量在坐标变换下不发生改变。	Length is invariant under rotation and translation. 长度在旋转和平移下保持不变。
scalar multiple	/ˈskeɪlə ˈmʌltɪpəl/	n. phr.		标量倍数	Vector \({\bf v}\) is a scalar multiple of \({\bf u}\) if \({\bf v}= k * {\bf u}\). 若 \({\bf v}= k * {\bf u}\)，则称向量 \({\bf v}\) 是 \({\bf u}\) 的数乘倍数。	The direction vector can be replaced by any scalar multiple. 方向向量可以替换为它的任意数乘倍数。
parameter	/pəˈræmɪtə(r)/	n.	parametric 参数化的； parametric representation 参数表示（参数方程）； parametric form 参数形式	参数	The curve depends on a parameter \(t\). 曲线依赖于参数\(t\)。	Changing the parameter alters the shape of the graph. 改变参数会改变图像形状。
geometrical interpretation	/ˌdʒiːəˈmetrɪkl ˌɪntəprəˈteɪʃn/	n. phr.		几何解释	We give a geometrical interpretation of the equation. 我们给出该方程的几何解释。	The geometrical interpretation of derivative is slope. 导数的几何解释是斜率。
radian	/ˈreɪdiən/	n.		弧度	One radian equals \(\frac{180}{\pi}\) degrees. \(1\)弧度约等于\(\frac{180}{\pi}\)度。	Angles in calculus are measured in radians. 微积分中角用弧度表示。
theta	/ˈθiːtə/	n.		\(\theta\) （角变量）	\(\theta\) represents an angle in polar coordinates. \(\theta\)表示极坐标中的角。	The angle \(\theta\) varies between \(0\) and \(2 \pi\). 角度\(\theta\)在\(0\)到\(2 \pi\)之间变化。
delta	/ˈdeltə/	n.		\(\delta\) （变化量）	We use \(\delta x\) in differentiation. 微分中使用\(\delta x\)。	Delta represents change in a variable. \(\delta\)表示变量的变化。
pi	/paɪ/	n.		圆周率 \(\pi\)	The circumference of a circle is \(2 \pi r\). 圆的周长是\(2 \pi r\)。	The value of \(\pi\) is approximately \(3.14159\). \(\pi\)的值约为\(3.14159\)。
orient	/ˈɔːrɪənt/	v.		标定方向	We orient the normal vector \({\bf n}\) outward for the closed surface. 我们为闭曲面将法向量 \({\bf n}\) 取朝外定向。	The right-hand rule helps orient the \(3D\) coordinate system. 右手定则可用来给三维坐标系确定定向。
clockwise	/ˈklɒkwaɪz/	adj./adv.	anticlockwise, counterclockwise 逆时针	顺时针	A clockwise rotation by \(90\) degrees maps \((1,0)\) to \((0,-1)\). 顺时针旋转 \(90\) 度将点 \((1,0)\) 映射为 \((0,-1)\)。	Draw the circular path in clockwise direction for line integral. 沿顺时针方向绘制圆周路径以计算曲线积分。
intersection	/ˌɪntəˈsekʃn/	n.	intersect v. 相交，交于； point of intersection 交点； common point 公共点； line of intersection 交线	交点，交集	The intersection of a circle and a line can give \(0, 1\) or \(2\) points. 圆和直线的交点可能有\(0\)个, \(1\)个或\(2\)个。	The intersection of the two planes is a straight line in \(3D\) space. 两个平面的交线是在三维空间中的一条直线。
Pythagorean	/ˌpaɪθæɡəˈriːən/	adj.	Pythagoras 毕达哥拉斯	毕达哥拉斯的	The distance formula comes from the Pythagorean theorem. 距离公式来源于勾股定理。	The Pythagorean formula is \(a^2+b^2=c^2\). 勾股公式是\(a^2+b^2=c^2\)。
Gaussian	/ˈɡaʊsiən/	adj.	Gauss 高斯	高斯的	We use Gaussian elimination to solve systems of linear equations. 我们用高斯消元法求解线性方程组。	Gaussian integers are numbers of the form \(a+bi\) where \(a,b \in \mathbb{Z}\). 高斯整数是形如\(a+bi\)且\(a, b\)属于整数集\(\mathbb{Z}\)的数。