To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second one. That means, keep the first fraction the way it is. Change the division to a multiplication. Flip the second fraction.
-8/3 / -2/6 = -8/3 * -6/2 = (8 * 6)/(3 * 2) = 8
Answer:
y = -2000x+30000
Step-by-step explanation:
Hi. So, you are just setting up a linear equation using slope intercept data. The slope being -2000 (because it is descending) and the intercept being 30,000 at t=0.
y = -2000x+30000
Answer:
8x
Step-by-step explanation:
find the orthogonal projection of v= [19,12,14,-17] onto the subspace W spanned by [ [ -4,-1,-1,3] ,[ 1,-4,4,3] ] proj w (v) = [
12345 [234]
<h2>
Answer:</h2>
Hence, we have:
![proj_W(v)=[\dfrac{464}{21},\dfrac{167}{21},\dfrac{71}{21},\dfrac{-131}{7}]](https://tex.z-dn.net/?f=proj_W%28v%29%3D%5B%5Cdfrac%7B464%7D%7B21%7D%2C%5Cdfrac%7B167%7D%7B21%7D%2C%5Cdfrac%7B71%7D%7B21%7D%2C%5Cdfrac%7B-131%7D%7B7%7D%5D)
<h2>
Step-by-step explanation:</h2>
By the orthogonal decomposition theorem we have:
The orthogonal projection of a vector v onto the subspace W=span{w,w'} is given by:
Here we have:
![v=[19,12,14,-17]\\\\w=[-4,-1,-1,3]\\\\w'=[1,-4,4,3]](https://tex.z-dn.net/?f=v%3D%5B19%2C12%2C14%2C-17%5D%5C%5C%5C%5Cw%3D%5B-4%2C-1%2C-1%2C3%5D%5C%5C%5C%5Cw%27%3D%5B1%2C-4%2C4%2C3%5D)
Now,
![v\cdot w=[19,12,14,-17]\cdot [-4,-1,-1,3]\\\\i.e.\\\\v\cdot w=19\times -4+12\times -1+14\times -1+-17\times 3\\\\i.e.\\\\v\cdot w=-76-12-14-51=-153](https://tex.z-dn.net/?f=v%5Ccdot%20w%3D%5B19%2C12%2C14%2C-17%5D%5Ccdot%20%5B-4%2C-1%2C-1%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%3D19%5Ctimes%20-4%2B12%5Ctimes%20-1%2B14%5Ctimes%20-1%2B-17%5Ctimes%203%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%3D-76-12-14-51%3D-153)
![w\cdot w=[-4,-1,-1,3]\cdot [-4,-1,-1,3]\\\\i.e.\\\\w\cdot w=(-4)^2+(-1)^2+(-1)^2+3^2\\\\i.e.\\\\w\cdot w=16+1+1+9\\\\i.e.\\\\w\cdot w=27](https://tex.z-dn.net/?f=w%5Ccdot%20w%3D%5B-4%2C-1%2C-1%2C3%5D%5Ccdot%20%5B-4%2C-1%2C-1%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%5Ccdot%20w%3D%28-4%29%5E2%2B%28-1%29%5E2%2B%28-1%29%5E2%2B3%5E2%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%5Ccdot%20w%3D16%2B1%2B1%2B9%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%5Ccdot%20w%3D27)
and
![v\cdot w'=[19,12,14,-17]\cdot [1,-4,4,3]\\\\i.e.\\\\v\cdot w'=19\times 1+12\times (-4)+14\times 4+(-17)\times 3\\\\i.e.\\\\v\cdot w'=19-48+56-51\\\\i.e.\\\\v\cdot w'=-24](https://tex.z-dn.net/?f=v%5Ccdot%20w%27%3D%5B19%2C12%2C14%2C-17%5D%5Ccdot%20%5B1%2C-4%2C4%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%27%3D19%5Ctimes%201%2B12%5Ctimes%20%28-4%29%2B14%5Ctimes%204%2B%28-17%29%5Ctimes%203%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%27%3D19-48%2B56-51%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%27%3D-24)
![w'\cdot w'=[1,-4,4,3]\cdot [1,-4,4,3]\\\\i.e.\\\\w'\cdot w'=(1)^2+(-4)^2+(4)^2+(3)^2\\\\i.e.\\\\w'\cdot w'=1+16+16+9\\\\i.e.\\\\w'\cdot w'=42](https://tex.z-dn.net/?f=w%27%5Ccdot%20w%27%3D%5B1%2C-4%2C4%2C3%5D%5Ccdot%20%5B1%2C-4%2C4%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%27%5Ccdot%20w%27%3D%281%29%5E2%2B%28-4%29%5E2%2B%284%29%5E2%2B%283%29%5E2%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%27%5Ccdot%20w%27%3D1%2B16%2B16%2B9%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%27%5Ccdot%20w%27%3D42)
Hence, we have:
![proj_W(v)=(\dfrac{-153}{27})[-4,-1,-1,3]+(\dfrac{-24}{42})[1,-4,4,3]\\\\i.e.\\\\proj_W(v)=\dfrac{-17}{3}[-4,-1,-1,3]+(\dfrac{-4}{7})[1,-4,4,3]\\\\i.e.\\\\proj_W(v)=[\dfrac{68}{3},\dfrac{17}{3},\dfrac{17}{3},-17]+[\dfrac{-4}{7},\dfrac{16}{7},\dfrac{-16}{7},\dfrac{-12}{7}]\\\\i.e.\\\\proj_W(v)=[\dfrac{464}{21},\dfrac{167}{21},\dfrac{71}{21},\dfrac{-131}{7}]](https://tex.z-dn.net/?f=proj_W%28v%29%3D%28%5Cdfrac%7B-153%7D%7B27%7D%29%5B-4%2C-1%2C-1%2C3%5D%2B%28%5Cdfrac%7B-24%7D%7B42%7D%29%5B1%2C-4%2C4%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cproj_W%28v%29%3D%5Cdfrac%7B-17%7D%7B3%7D%5B-4%2C-1%2C-1%2C3%5D%2B%28%5Cdfrac%7B-4%7D%7B7%7D%29%5B1%2C-4%2C4%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cproj_W%28v%29%3D%5B%5Cdfrac%7B68%7D%7B3%7D%2C%5Cdfrac%7B17%7D%7B3%7D%2C%5Cdfrac%7B17%7D%7B3%7D%2C-17%5D%2B%5B%5Cdfrac%7B-4%7D%7B7%7D%2C%5Cdfrac%7B16%7D%7B7%7D%2C%5Cdfrac%7B-16%7D%7B7%7D%2C%5Cdfrac%7B-12%7D%7B7%7D%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cproj_W%28v%29%3D%5B%5Cdfrac%7B464%7D%7B21%7D%2C%5Cdfrac%7B167%7D%7B21%7D%2C%5Cdfrac%7B71%7D%7B21%7D%2C%5Cdfrac%7B-131%7D%7B7%7D%5D)
The "middle" of a sorted list of numbers. To find the Median, place the numbers in value order and find the middle number. ... The middle number is 15, so the median is 15. (When there are two middle numbers we average them.)
