Векторы и преобразовательная геометрия
Pearson Edexcel · International GCSE · Mathematics A Тема 5 3:30 Английское озвучивание · Английский + китайские субтитры (встроенные)
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A column vector goes across, then up, and a scalar multiple stretches both parts.
列向量先写横向再写纵向，数乘把两个分量同时放大。
Its magnitude comes from Pythagoras.
它的大小由勾股定理求得。
Vectors add tip to tail along any route, and reversing a vector negates it.
向量沿任意路线首尾相接相加，反向就取负。
Each transformation needs its full description: a mirror line, a centre with angle and direction, a vector, or a centre with a scale factor.
每种变换都要完整描述：镜像线；旋转中心、角度和方向；平移向量；或位似中心和比例因子。
A vector has a size and a direction.
向量有大小和方向。
Multiplying by a scalar stretches it, and a minus sign reverses it.
乘以一个数量会把它拉长，负号会使它反向。
Column vectors add component by component: 2, 5 plus minus 1, 3 is 1, 8.
列向量逐分量相加：(2, 5) 加 (负 1, 3) 等于 (1, 8)。
On the grid, the second vector starts where the first one ends, tip to tail, and the resultant runs straight from the start to the finish.
在网格上，第二个向量 从第一个的终点出发，首尾相接，合向量从起点直接指向终点。
Its length is the modulus, by Pythagoras: root of 1 plus 64, which is root 65, about 8.06.
它的长度是模，用勾股定理：根号下 1 加 64，即根号 65，约等于 8.06。
OABC is a parallelogram, with OA is a and OC is c.
OABC 是平行四边形，OA = a，OC = c。
Diagonal OB is a plus c, and M is the midpoint of OB, so OM is a half of a plus c.
对角线 OB = a + c，M 是 OB 的中点，所以 OM = ½(a + c)。
From A, AM is minus a plus OM, which is a half of c minus a.
从 A 出发，AM = −a + OM = ½(c − a)。
But the other diagonal AC is c minus a.
而另一条对角线 AC = c − a。
So AM is half of AC: M is the midpoint of both diagonals.
所以 AM 是 AC 的一半：M 是两条对角线共同的中点。
In triangle OAB, OA is a and OB is b.
在三角形 OAB 中，OA = a，OB = b。
P lies on AB one third of the way from A, so OP is two thirds a plus one third b.
P 在 AB 上，距 A 三分之一，所以 OP = ⅔a + ⅓b。
Take M the midpoint of OB.
取 M 为 OB 的中点。
OP and AM meet at Q.
OP 与 AM 交于 Q。
Write OQ as k times OP, and again as a plus lambda times AM.
把 OQ 写成 k·OP，再写成 a + λ·AM。
Equate the coefficients of a and of b: k is three quarters, so OQ to OP is 3 to 4.
比较 a 和 b 的系数：k = ¾，所以 OQ : OP = 3 : 4。
Every isometry keeps lengths and angles, so the image is congruent.
每种等距变换都保持长度和角度，所以像是全等的。
A rotation needs its centre, angle and direction.
旋转要说明旋转中心、角度和方向。
Rotating 90 degrees anticlockwise about the origin sends x, y to minus y, x: the triangle at 1, 1, then 3, 1, then 1, 2 lands at minus 1, 1, then minus 1, 3, then minus 2, 1.
绕原点逆时针旋转 90 度，把 (x, y) 变成 (负 y, x)：顶点 为 (1, 1)、(3, 1)、(1, 2) 的三角形落到 (负 1, 1)、(负 1, 3)、(负 2, 1)。
A reflection needs its mirror line: in the x axis, x, y goes to x, minus y.
反射要说明对称轴：关于 x 轴反射，(x, y) 变成 (x, 负 y)。
A translation is a column vector.
平移 用一个列向量表示。
An enlargement needs a centre and a scale factor.
放大要说明中心和比例因子。
About the origin, scale factor 2 sends x, y to 2 x, 2 y: the triangle at 1, 1, then 2, 1, then 1, 2 becomes 2, 2, then 4, 2, then 2, 4.
以原点为中心、比例因子为 2，把 (x, y) 变成 (2x, 2y)：顶点为 (1, 1)、(2, 1)、(1, 2) 的三角形变成 (2, 2)、 (4, 2)、(2, 4)。
Rays from the centre through each corner pass through the matching image corner, which confirms the enlargement.
从中心经过每个顶点的射线都经过对应的像顶点，这就确认了放大。
Lengths double but angles stay the same, so the image is similar, not congruent.
长度加倍而角度不变，所以像是相似的，不是全等的。
Check yourself.
自我检测。
The magnitude of 3, minus 5 is root 34, about 5.83.
(3, −5) 的大小是 √34，约 5.83。
Reversing b minus a gives a minus b.
b − a 反向得 a − b。
2a is parallel to a and twice as long.
2a 与 a 平行，长度是两倍。
A rotation needs a centre, an angle and a direction.
旋转需要中心、角度和方向。
And an enlargement keeps the angles but not the lengths: the image is similar.
位似变换保持角度但不保持长度：像与原图相似。