Slopes of parallel and perpendicular lines Video transcript We're asked what is the equation of line B?
It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive? Not according to s mathematician Benjamin Peirce: It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.
Argh, this attitude makes my blood boil! Formulas are not magical spells to be memorized: Euler's formula describes two equivalent ways to move in a circle.
This stunning equation is about spinning around? Yes -- and we can understand it by building on a few analogies: Starting at the number 1, see multiplication as a transformation that changes the number: If they can't think it through, Euler's formula is still a magic spell to them.
While writing, I thought a companion video might help explain the ideas more clearly: It follows the post; watch together, or at your leisure. Euler's formula is the latter: If we examine circular motion using trig, and travel x radians: The analogy "complex numbers are 2-dimensional" helps us interpret a single complex number as a position on a circle.
Now let's figure out how the e side of the equation accomplishes it. What is Imaginary Growth? Combining x- and y- coordinates into a complex number is tricky, but manageable.
But what does an imaginary exponent mean? Let's step back a bit. When I see 34, I think of it like this: Regular growth is simple: Imaginary growth is different: It's like a jet engine that was strapped on sideways -- instead of going forward, we start pushing at 90 degrees.
The neat thing about a constant orthogonal perpendicular push is that it doesn't speed you up or slow you down -- it rotates you! Taking any number and multiplying by i will not change its magnitude, just the direction it points.
Intuitively, here's how I see continuous imaginary growth rate: I wondered that too. Regular growth compounds in our original direction, so we go 1, 2, 4, 8, 16, multiplying 2x each time and staying in the real numbers. We can consider this eln 2xwhich means grow instantly at a rate of ln 2 for "x" seconds.
And hey -- if our growth rate was twice as fast, 2ln 2 vs ln 2it would look the same as growing for twice as long 2x vs x.
The magic of e lets us swap rate and time; 2 seconds at ln 2 is the same growth as 1 second at 2ln 2. Now, imagine we have some purely imaginary growth rate Ri that rotates us until we reach i, or 90 degrees upward.
What happens if we double that rate to 2Ri, will we spin off the circle? Having a rate of 2Ri means we just spin twice as fast, or alternatively, spin at a rate of R for twice as long, but we're staying on the circle.
Rotating twice as long means we're now facing degrees.
Once we realize that some exponential growth rate can take us from 1 to i, increasing that rate just spins us more. We'll never escape the circle.
But let's not get fancy: Euler's formula, eix, is about the purely imaginary growth that keeps us on the circle more later. Why use ex, aren't we rotating the number 1? When we write e we're capturing that entire process in a single number -- e represents all the whole rigmarole of continuous growth.Find the Equation of a Line Parallel or Perpendicular to Another Line As we have seen when finding the equation of a line given two points, there are three steps required to find the answer.
These three steps are: Step 1: Find the slope of the line. Step 2: Use the slope to find the y-intercept. Step 3: Use steps 1 and 2 to write the answer. You must know the structure of a straight-line equation before you can write equations for parallel or perpendicular lines.
The standard form of the equation is "y = mx + b," in which "m" is the slope of the line and "b" is the point where the line crosses the y-axis. It's called the point-slope formula (Duh!) You are going to use this a LOT!
Luckily, it's pretty easy -- let's just do one: Let's find the equation of the line that passes through the point. Find the Equation of a Line Parallel or Perpendicular to Another Line – Notes Page 3 of 4 Example 4: Find the equation of a line passing through the point (3, –4) perpendicular to the line 8x + 6y = Step 1: Find the slope of the line.
To find the slope of the given line . The equation of line p is y= 1/4x +6. If line k contains the point (3,5) and is perpendicular to line p, find the equation of line k.
Write the equation of a line perpendicular to y=-5x+6 thru (6,-2) When one line is perpendicular to another line their slopes are the negative reciprocal of each other.
In other words, if you multiply the two slopes together you get Write an equation of the line perpendicular to the graph of x=3 and passing through D(4, -1).