Transformations and vectors
IGCSE Mathematics Topic 7 7:53 English narration · English + 中文 subtitles burned in
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Transcript
Look at this wall.
看看这面墙。
It seems complicated — hundreds of shapes locking together.
它看起来很复杂——上百个形状严丝合缝地拼在一起。
But nobody drew every piece.
但没有人去画每一块。
Someone drew one small shape, then moved it again and again. Slide it. Flip it.
有人只画了一个小形状，然后一次又一次地移动它：平移、翻转、旋转、放大。
Turn it. Make it bigger. That is today's topic: four moves, called transformations, and the arrows that describe them, called vectors.
这就是今天的内容：四种移动，叫做变换；以及描述它们的箭头，叫做向量。
One flag, four moves.
一面小旗，四种移动。
A translation slides it across.
平移让它整体滑动。
A reflection flips it over a mirror line.
反射让它翻过一条对称轴。
A rotation turns it about a fixed point.
旋转让它绕着一个定点转动。
An enlargement grows it away from a centre.
放大让它从一个中心向外变大。
Only the enlargement changes the size.
只有放大会改变大小。
Exam questions almost always say: describe fully the single transformation.
考题几乎总是这样问：完整描述这个变换。
That means name the type, then give every detail.
意思是先说出类型，再给出所有细节。
For a reflection, give the equation of the mirror line.
反射，要给出对称轴的方程。
For a rotation, give three things: the centre, the angle, and the direction.
旋转，要给出三样：中心、角度和方向。
For an enlargement, give the centre and the scale factor.
放大，要给出中心和比例因子。
For a translation, give the column vector.
平移，要给出列向量。
Miss one detail and you lose the mark.
少写一个细节就丢分。
A reflection flips the shape over a mirror line.
反射把图形翻过一条对称轴。
Every point and its image sit the same distance from that line, on opposite sides, and the join between them crosses the line at a right angle.
每个点和它的像到这条线的距离相同，分别在两侧， 连线与这条线垂直。
A point already on the line does not move.
已经落在线上的点不动。
Now the coordinate rule.
现在看坐标规律。
Reflect in the upright axis and only the first coordinate changes sign. Reflect in the flat axis and only the second changes sign.
对竖直轴作反射，只有第一个坐标变号； 对水平轴作反射，只有第二个坐标变号。
Always the opposite letter to the axis named.
永远是与所给坐标轴相反的那个字母。
A rotation turns the shape about a fixed point called the centre.
旋转让图形绕着一个定点转动，这个定点叫做旋转中心。
Here the centre is the origin and the turn is ninety degrees anticlockwise.
这里中心是原点，转动是逆时针九十度。
To describe it you need all three: the centre, the angle, and the direction.
要描述它，三样都要有：中心、角度和方向。
At this level the angle is always a multiple of ninety degrees, and for a half turn the direction does not matter.
在这一阶段角度总是九十度的倍数， 而转半圈时方向无所谓。
Mark the centre with a small cross first — that is the part students forget.
先用一个小十字标出中心——这正是同学最容易漏掉的部分。
An enlargement changes the size.
放大会改变大小。
Every distance from the centre is multiplied by the scale factor.
每一段到中心的距离都乘以比例因子。
The dashed rays all start at the centre and pass through matching corners, so the image sits twice as far out.
这些虚线射线都从中心出发，穿过对应的顶点，所以像落在两倍远的地方。
To describe it, give the centre and the scale factor.
要描述它，给出中心和比例因子。
A factor between zero and one makes the shape smaller — still an enlargement.
比例因子在零和一之间时图形变小——仍然叫做放大。
On the Extended paper it can be negative, putting the image on the other side of the centre, upside down.
在扩展卷中它还可以是负数，那会把像放到中心的另一侧，并且上下颠倒。
A translation slides the shape. No turning, no flipping, no change of size.
平移让图形滑动：不转动、不翻转、大小不变。
Every point moves by exactly the same amount, which is why the three arrows here are parallel and the same length.
每个点移动的量完全相同，所以图中这三支箭头平行且等长。
We write that amount as a column vector: the top number is how far across, the bottom is how far up.
我们把这个量写成列向量：上面的数是横向走多远，下面的是纵向走多高。
A negative top number means left, a negative bottom number means down.
上面的数为负表示向左，下面的数为负表示向下。
The exam often gives you coordinates instead of a picture, so learn the point rules.
考试常常给你坐标而不是图，所以要记住点的规律。
Rotate the point three, one by ninety degrees anticlockwise about the origin: swap the numbers, then change the sign of the new first one, giving minus one, three.
把点三、一绕原点逆时针旋转九十度：把两个数交换，再把新的第一个数变号，得到负一、三。
Enlarge the point one, two from the origin with scale factor two: multiply both coordinates, giving two, four.
把点一、二以原点为中心按比例因子二放大：两个坐标都相乘，得到二、四。
Translate the point five, three by two left and four up, giving three, seven.
把点五、三向左二、向上四平移，得到三、七。
Now vectors.
现在讲向量。
Wind is a good example: it has a size — how strong it blows — and a direction — the way it blows.
风就是个好例子：它有大小——吹得多强；也有方向——往哪边吹。
Any quantity with both size and direction is a vector.
任何既有大小又有方向的量都是向量。
We write one as a column vector, across on top and up below; or as an arrow over two letters, meaning from the first point to the second; or as one bold letter.
我们把它写成列向量，上面是横向、下面是纵向； 也可以在两个字母上加箭头，表示从第一个点到第二个点；还可以写成一个粗体字母。
Everything from here is Extended paper only.
从这里开始都只属于扩展卷。
To add two vectors, draw them tip to tail: the second arrow starts where the first one ends.
要把两个向量相加，就首尾相接地画：第二支箭头从第一支结束的地方开始。
The answer, called the resultant, runs from the very start to the very end.
得到的结果叫做合向量，它从最开始一直指到最末端。
You can also complete the parallelogram and take the diagonal.
你也可以补成一个平行四边形，取它的对角线。
Same arrow, every time.
每次都是同一支箭头。
On paper it is even easier.
在纸上算更简单。
Add or subtract by working on the top numbers, then the bottom numbers, and never mix them.
加减时分别处理上面的数和下面的数，绝不混在一起。
Take a as three across and one up, and b as two across and four down.
设 a 是横向三、纵向上一，b 是横向二、纵向下四。
Add the tops: three plus two is five.
上面相加：三加二等于五。
Add the bottoms: one plus minus four is minus three.
下面相加：一加负四等于负三。
So a plus b is five across and three down.
所以 a 加 b 是横向五、纵向下三。
To multiply by a scalar — an ordinary number — multiply both parts: three times a is nine across and three up.
要乘一个标量——也就是普通的数——就把两部分都乘：三倍的 a 是横向九、纵向上三。
How long is a vector?
一个向量有多长？
That length is its magnitude, written between two upright lines.
这个长度叫做它的模，写在两条竖线之间。
In the triangle, the across part is one side, the up part is the other, and the vector is the sloping side.
在这个三角形里，横向的部分是一条直角边，纵向的部分是另一条，向量本身是斜边。
So Pythagoras gives the length: square the across part, square the up part, add, then take the square root.
所以用勾股定理求长度：横向部分平方，纵向部分平方，相加，再开平方根。
For three across and four up, nine plus sixteen is twenty-five, and its square root is five.
横三纵四：九加十六等于二十五，它的平方根是五。
A magnitude is never negative.
模永远不会是负数。
A vector can be drawn as a directed line segment — an arrow.
向量可以画成有向线段——一支箭头。
The position vector of a point is the arrow from the origin to that point.
一个点的位置向量，就是从原点指向那个点的箭头。
Here a is the position vector of A, and b is the position vector of B.
这里 a 是 A 的位置向量，b 是 B 的位置向量。
So what is the arrow from A to B?
那么从 A 到 B 的箭头是什么？
You cannot jump there, so travel backwards along a to the origin, then forwards along b.
你不能直接跳过去，所以先沿着 a 倒回原点，再沿着 b 前进。
Backwards along a is minus a, so the vector from A to B is b minus a: end minus start.
沿 a 倒回就是负的 a， 于是从 A 到 B 的向量是 b 减 a：终点减起点。
Almost every vector geometry question starts there.
几乎每道向量几何题都从这里开始。
Two tools come up again and again.
有两个工具反复出现。
First, parallel.
第一，平行。
Two vectors are parallel when one is simply a number times the other.
当一个向量是另一个向量的数倍时，它们平行。
If one arrow is two times another, they point the same way and one is twice as long, so they are parallel.
如果一支箭头是另一支的两倍，它们方向相同、长度是两倍，所以平行。
Second, collinear — three points on one straight line.
第二，共线——三个点在同一条直线上。
Show that two of the vectors between them are parallel, and that they share a point.
证明它们之间的两个向量平行，并且有一个公共点。
Then the three points are collinear.
那么这三个点就共线。
You can also express any vector in terms of two coplanar vectors.
你也可以用两个共面向量来表示任意向量。
The classic question.
这是一道经典题。
O is the origin, the position vector of A is a, and the position vector of B is b.
O 是原点，A 的位置向量是 a，B 的位置向量是 b。
M is the midpoint of the line joining A and B.
M 是连接 A 和 B 的线段的中点。
Find the vector from O to M in terms of a and b. Pause and try it.
用 a 和 b 表示从 O 到 M 的向量。
Step one: the vector from A to B is b minus a — end minus start.
先暂停，自己试一试。 第一步：从 A 到 B 的向量是 b 减 a——终点减起点。
Step two: M is halfway along, so the vector from A to M is one half of b minus a.
第二步：M 在正中间，所以从 A 到 M 的向量是二分之一的 b 减 a。
Step three: to reach M from the origin, travel along a first, then along A to M, giving a plus one half of b minus a. Now tidy that up.
第三步：要从原点到达 M，先沿着 a 走，再沿着 A 到 M 走，得到 a 加二分之一的 b 减 a。
It becomes one half a plus one half b, which is one half of a plus b — one of the most common answers in the whole topic.
现在整理一下：它变成二分之一 a 加二分之一 b，也就是二分之一的 a 加 b—— 这是整个专题里最常见的答案之一。
Four marks students throw away.
四个同学常丢的分。
One: describe fully means the type plus every detail — a rotation needs centre, angle and direction.
第一：完整描述的意思是类型加上每一个细节—— 旋转需要中心、角度和方向。
Two: a fractional scale factor is still an enlargement, and a negative one turns the image upside down through the centre.
第二：分数比例因子仍然是放大， 而负的比例因子会让像通过中心上下颠倒。
Three: add vectors top to top and bottom to bottom; never mix the rows.
第三：向量相加要上对上、下对下，绝不混行。
Four: the vector from A to B is b minus a — end minus start.
第四：从 A 到 B 的向量是 b 减 a——终点减起点。
Get those right and this topic is yours.
这几点做对，这个专题就是你的了。