How to use Algebra to find parallel and perpendicular lines.
Parallels Desktop® for Mac is the fastest, easiest, and most powerful application for running Windows® on a Mac®—without rebooting. Brought to you by the world-class developers of the #1-rated Mac virtualization software. Note: It is not recommended that existing Parallels Desktop® for Mac users m. The words analogous and similar are common synonyms of parallel. While all three words mean 'closely resembling each other,' parallel suggests a marked likeness in the development of two things. The parallel careers of two movie stars When can analogous be used instead of parallel? Parallel to y = 2x + 1; and passes though the point (5,4) The slope of y=2x+1 is: 2. The parallel line needs to have the same slope of 2. We can solve it using the 'point-slope' equation of a line: y − y 1 = 2(x − x 1) And then put in the point (5,4): y − 4 = 2(x − 5) And that answer is OK, but let's also put it in y = mx + b form: y. GNU Parallel默认把一行做为一个参数：使用 做为参数定界符。可以使用 -d 改变： 21:25 sxuan@hulab $ parallel -d b echo:::: a.txt a c d e 1.4 提前结束和跳过空行. GNU Parallel支持通过-E参数指定一个值做为结束标志： 21:26 sxuan@hulab $ parallel -E stop echo::: A B stop C D A B.
Parallel Lines
How do we know when two lines are parallel?
The slope is the value m in the equation of a line: y = mx + b |
Example:
Find the equation of the line that is:
- parallel to y = 2x + 1
- and passes though the point (5,4)
The slope of y=2x+1 is: 2
Hamamatsu photonics k.k port devices driver. The parallel line needs to have the same slope of 2.
We can solve it using the 'point-slope' equation of a line:
y − y1 = 2(x − x1)
And then put in the point (5,4):
y − 4 = 2(x − 5)
And that answer is OK, but let's also put it in y = mx + b form:
y − 4 = 2x − 10
y = 2x − 6
Vertical Lines
But this does not work for vertical lines .. I explain why at the end.
Not The Same Line
Be careful! They may be the same line (but with a different equation), and so are not parallel.
How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope:
Example: is y = 3x + 2 parallel to y − 2 = 3x ?
For y = 3x + 2: the slope is 3, and y-intercept is 2
For y − 2 = 3x: the slope is 3, and y-intercept is 2
In fact they are the same line and so are not parallel
Perpendicular Lines
Two lines are Perpendicular when they meet at a right angle (90°).
To find a perpendicular slope:
When one line has a slope of m, a perpendicular line has a slope of −1m
In other words the negative reciprocal
Example:
Find the equation of the line that is
- perpendicular to y = −4x + 10
- and passes though the point (7,2)
The slope of y=−4x+10 is: −4
The negative reciprocal of that slope is:
m = −1−4 = 14
So the perpendicular line will have a slope of 1/4:
y − y1 = (1/4)(x − x1)
And now put in the point (7,2):
y − 2 = (1/4)(x − 7)
And that answer is OK, but let's also put it in 'y=mx+b' form:
y − 2 = x/4 − 7/4
y = x/4 + 1/4
Quick Check of Perpendicular
When we multiply a slope m by its perpendicular slope −1m we get simply −1.
So to quickly check if two lines are perpendicular:
Like this:
Parallelogram
Are these two lines perpendicular?
When we multiply the two slopes we get:
2 × (−0.5) = −1
Yes, we got −1, so they are perpendicular.
Vertical Lines
The previous methods work nicely except for a vertical line:
In this case the gradient is undefined (as we cannot divide by 0):
m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined
So just rely on the fact that:
- a vertical line is parallel to another vertical line.
- a vertical line is perpendicular to a horizontal line (and vice versa).
Summary
- parallel lines: same slope
- perpendicular lines: negative reciprocal slope (−1/m)
1540s, in geometry, of lines, 'lying in the same plane but never meeting in either direction;' of planes, 'never meeting, however far extended;' from French parallèle (16c.) and directly from Latin parallelus, from Greek parallēlos 'parallel,' from para allēlois 'beside one another,' from para- 'beside' (see para- (1)) + allēlois 'each other,' from allos 'other' (from PIE root *al- 'beyond'). Figurative sense of 'having the same direction, tendency, or course' is from c. 1600.
As a noun from 1550s, 'a line parallel to another line.' Meanings 'a comparison made by placing things side by side' and 'thing equal to or resembling another in all particulars' are from 1590s. Parallel bars as gymnastics apparatus is recorded from 1868.
parallel (v.)
1590s, transitive, 'place in position parallel to something else,' from parallel (n.). Ephone modems driver download for windows 10. Meaning 'make closely similar to something else' is from 1620s; intransitive sense of 'be like or equal, agree' is from 1620s.
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