Functions Involving Parameters, Vectors, and Matrices
AP Precalculus Topic 4 8:04 English narration · English + 中文 subtitles burned in
Chapters
Transcript
Look at this roller coaster.
看看这辆过山车。
At every moment the car has two facts: where it is, and which way it is going.
每一刻，车都有两个事实：它在哪里，以及它朝哪个方向走。
One equation cannot hold both.
一个方程装不下这两者。
This planet runs a closed path, then returns to the start.
这颗行星沿着一条闭合的路径运行，又回到起点。
For motion like that, we need a new tool.
要描述这样的运动，我们需要一个新工具。
A circle is not a function of x.
圆不是 x 的函数。
Try the vertical line test: one x value gives two y values, so no single function can draw it.
试试垂线检验：一个 x 值对应两个 y 值，所以没有哪个函数能画出它。
Now give the point a clock — let a third variable run forward like time.
现在给这个点一只钟——让第三个变量像时间一样向前走。
At each moment it names one x and one y.
每一刻，它都给出一个 x 和一个 y。
A path built this way may loop, cross itself, or come back.
用这种方式画出的路径可以绕圈、自我相交，也可以原路返回。
That third variable is the parameter, and we usually call it t.
这第三个变量就是参数，我们通常记作 t。
A parametric function gives x as a function of t, and y as a function of t, at the same time.
参数函数同时给出两件事： x 是 t 的函数，y 也是 t 的函数。
Because the coordinates are separate, the curve need not pass the vertical line test.
因为两个坐标是分开的，这条曲线不必通过垂线检验。
The domain of t matters too: it sets where the path starts, stops, and how often it goes round.
t 的定义域也很重要：它决定路径从哪里开始、在哪里结束，以及绕了几圈。
Read t as time, and a parametric function becomes motion in the plane.
把 t 读作时间，参数函数就变成了平面上的运动。
Watch the two balls. One is dropped; the other is launched sideways at the same moment.
看这两个球： 一个直接下落，另一个在同一瞬间被水平抛出。
They share every height and land together.
它们每一刻的高度都相同，也同时落地。
The sideways equation and the falling equation run independently, and together they draw the path.
水平方向的方程和竖直方向的方程各自独立，合在一起就画出了这条路径。
The top of an arc is where the vertical motion turns.
弧线的最高点，就是竖直运动掉头的地方。
A drone flies so that x is one plus two t, and y is four minus t, with t from zero to three.
一架无人机的位置是：x 等于一加二 t，y 等于四减 t，t 从零到三。
Where does it start and end?
它从哪里出发、到哪里结束？
At t equal to zero it is at one, four; at t equal to three, at seven, one.
t 等于零时在一、四；t 等于三时在七、一。
Which way does it travel?
它朝哪个方向走？
x grows and y falls, so it moves right and down.
x 在增大，y 在减小，所以它向右下方移动。
Now the average rate of change: x gains two per unit of t, y loses one.
再看平均变化率：t 每增加一个单位，x 增加二，y 减少一。
Divide the vertical rate by the horizontal rate: the slope is minus zero point five.
用竖直方向的变化率除以水平方向的变化率：斜率是负零点五。
Two rates, one slope.
两个变化率，一个斜率。
Three parametrizations are worth memorising.
有三种参数化值得记住。
For a circle about a centre, take cosine for x and sine for y, each times the radius; one full turn of t sweeps the circle.
对于以某点为圆心的圆，x 取余弦、y 取正弦，各自乘以半径； t 转一整圈，就扫过整个圆。
An ellipse is almost the same: a different radius across and up.
椭圆几乎一样：横向和纵向用不同的半径。
A line is even simpler: start at a point, then add t lots of a direction.
直线更简单：从一个点出发，再加上 t 倍的方向。
And an ordinary function needs one move: let x be t.
而普通函数只需一步：让 x 等于 t。
An equation can tie x and y together without solving for either.
一个方程可以把 x 和 y 绑在一起，却不解出其中任何一个。
That is an implicit equation, and the circle here is the classic case.
这就是隐式方程， 这里的圆就是最经典的例子。
The conic sections — circle, ellipse, hyperbola and parabola — are all implicit equations of this kind.
圆锥曲线——圆、椭圆、双曲线和抛物线—— 都属于这一类隐式方程。
Many implicit curves can be parametrized — rewritten with a clock t.
许多隐式曲线可以参数化——用一个时钟 t 重写。
To go back to a parameter: cosine squared plus sine squared is always one, so cosine and sine fit a circle.
要回到参数：余弦的平方加正弦的平方永远等于一， 所以余弦和正弦正好适合圆。
Why are they called sections?
为什么叫圆锥曲线？
They are one cone, cut at four angles. Cut straight across for a circle.
因为它们是同一个圆锥，用四种角度切出来的。
Tilt a little for an ellipse. Tilt until the cut runs parallel to the cone's own edge, for a parabola. Steeper still, and the cut meets both cones — a hyperbola, in two branches.
横着切，得到圆；稍微倾斜，得到椭圆；一直斜到切面与圆锥的母线平行，得到抛物线； 再陡一些，切面同时切到上下两个锥面——那就是双曲线，有两支。
The second big idea: vectors.
第二个大概念：向量。
A vector carries two things — a size, called the magnitude, and a direction.
向量带着两样东西——大小，我们称为模长，以及方向。
Watch the arrow turn.
看这个箭头转动。
At every angle it splits into a horizontal part and a vertical part; those two numbers are its components.
在每一个角度，它都分解成水平的一部分和竖直的一部分； 这两个数就是它的分量。
The magnitude comes from Pythagoras: square both components, add, then take the square root.
模长来自勾股定理：把两个分量分别平方，相加，再开平方根。
Adding vectors works in two matching ways.
向量相加有两种彼此对应的做法。
On paper, put the tail of the second arrow at the head of the first; the sum runs from the first tail to the last head.
在纸上，把第二个箭头的尾端接到第一个箭头的头端， 从最初的尾端到最后的头端，那一支箭头就是和。
In numbers, just add the matching components.
用数字算，就把对应的分量相加。
Multiplying by a number is the same pattern: the arrow keeps its direction but changes length — and a negative number turns it the other way.
乘以一个数也是同样的规律：箭头方向不变，长度改变——如果这个数是负的，箭头就掉头指向相反方向。
Take the vectors three, four and minus four, three.
取向量三、四，和向量负四、三。
First the sum: add the matching components, giving minus one and seven.
先求和：把对应的分量相加，得到负一和七。
Next the magnitude of the first vector: nine plus sixteen is twenty-five, so it is five.
再求第一个向量的模长：九加十六等于二十五，所以模长是五。
Then the unit vector: same direction, length one, divide each component by five.
接着是单位向量：方向相同，长度为一，把每个分量都除以五。
Last, the dot product: multiply the matching components and add. That gives zero — and a dot product of zero always means the two vectors are perpendicular.
最后是点积：把对应的分量相乘再相加，结果是零—— 而点积为零永远意味着这两个向量互相垂直。
Look at the picture: a right angle.
看这幅图：一个直角。
A vector-valued function packs all of this into one object.
向量值函数把这一切打包成一个对象。
For every t it returns a position vector — an arrow from the origin to where the point is.
对每一个 t，它返回一个位置向量—— 一支从原点指向该点所在位置的箭头。
This picture is a straight line written that way: start at the arrow a, then slide t lots of the direction arrow b.
这幅图就是用这种方式写出的一条直线： 从箭头 a 出发，再沿着方向箭头 b 滑动 t 倍。
At t equal to two, two steps along.
t 等于二时，就走了两步。
A matrix is a rectangular array of numbers in rows and columns.
矩阵就是由数字排成行和列的长方形阵列。
Add two of them entry by entry.
两个矩阵相加，把对应位置的数相加即可。
Multiplying is the interesting one: take the first row of the left matrix and the first column of the right, multiply pair by pair, and add — that gives the top-left entry.
相乘就有意思多了：取左边矩阵的第一行和右边矩阵的第一列，对应相乘再相加—— 这就得到左上角那个数。
It only works when the number of columns of the first matches the rows of the second.
只有当第一个矩阵的列数等于第二个矩阵的行数时，乘法才行得通。
And one matrix changes nothing: the identity matrix.
还有一个矩阵什么也不改变：单位矩阵。
Every two-by-two matrix carries one key number: its determinant.
每一个二阶矩阵都带着一个关键的数：它的行列式。
To find it, multiply the main diagonal, then subtract the product of the other two.
求法是：把主对角线上的两数相乘， 再减去另外两数的乘积。
Three times four is twelve, one times two is two, so twelve minus two is ten.
三乘四等于十二，一乘二等于二，所以十二减二等于十。
Because ten is not zero, this matrix has an inverse.
因为十不等于零，这个矩阵有逆矩阵。
Then swap the two numbers on the main diagonal, change the sign of the other two, and divide by the determinant.
接着把主对角线上的两个数对调， 另外两个数变号，最后全部除以行列式。
A matrix times its inverse gives the identity matrix.
矩阵乘以它的逆，得到单位矩阵。
Feed a matrix a vector and it hands back another vector, so a matrix is a linear transformation on the whole plane.
给矩阵一个向量，它就还回另一个向量，所以矩阵是整个平面上的一个线性变换。
The columns tell you everything: each says where one of the two unit arrows lands.
列告诉你一切：每一列说明两个单位箭头之一落在哪里。
Now watch the unit square.
现在看这个单位正方形。
It becomes a parallelogram, and its new area is exactly the determinant.
它变成了平行四边形，而它的新面积正好等于行列式。
A negative determinant means the plane has been flipped.
行列式为负，说明平面被翻转了过来。
One line says it all: a unit square goes in, a parallelogram comes out, and its area is the determinant.
一句话就说清了：一个单位正方形进去，一个平行四边形出来， 而这个平行四边形的面积就是行列式。
Take the matrix that stretches by two across and three up.
取那个横向拉伸二倍、纵向拉伸三倍的矩阵。
One, one lands on two, three; the determinant is six, so every area is multiplied by six.
一、一被映射到二、三；行列式是六，所以每一块面积都乘以六。
If the determinant were zero, the square would be squashed flat onto a line — exactly when no inverse exists.
如果行列式等于零，正方形就会被压扁成一条线——那正是逆矩阵不存在的时候。
Two last ideas.
最后还有两点。
First, because a matrix is a function, one transformation after another is the same as multiplying their matrices — and order matters, just as for functions.
第一，因为矩阵是一个函数，先做一个变换再做另一个变换， 就等于把它们的矩阵相乘——而且顺序很重要，就像函数复合一样。
Second, matrices model change over time.
第二，矩阵可以为随时间变化的系统建模。
Write today's populations as a state vector, multiply by a transition matrix, and you get next year's.
把今天的人口写成一个状态向量， 乘以一个转移矩阵，就得到明年的状态。
Matrix powers predict the long run.
矩阵的幂可以预测长期的走向。
Three marks students lose.
三个学生常失的分。
First, with a parametric function, always follow the direction as t increases — the same curve drawn backwards is a different answer.
第一，遇到参数函数，一定要跟住 t 增大时的运动方向—— 同一条曲线反过来画，就是不同的答案。
Second, for a magnitude: square, add, square root; do not skip the root.
第二，求模长时：平方、相加、开平方根， 别漏掉最后开方那一步。
Third, mind the minus sign in the determinant — a subtraction, not an addition.
第三，注意行列式中间的负号——那是减法，不是加法。
One honest note about this unit: it is not on the AP exam, which tests units one to three.
关于这一单元还有一点要老实说：它不在 AP 考试范围内，考试只考第一到第三单元。
It is here because it is the bridge into calculus.
它出现在这里，是因为它是通往微积分的桥梁。