# math

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a 25 foot ladder leans against the wallit touches the wall 24 feet off the ground if you slide the base of the ladder back 10 feet off the ground how far does it slide down the wall?

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a little vector analysis makes this relatively easy. Find two points B and C on the line. Let A=(1,2,2)

Form vectors AB and AC. These two vectors lie in the desired plane. Let n be the normal to the plane.

n = AB × AC
n•(v-A) = 0 is the desired plane.

So, pick any two points on the line, say B=(0,2,0) and C=(1,4,0)

AB = -i -2k
AC = 2j -2k

n =
| i j k |
|-1 0 -2 |
| 0 2 -2 |
= 4i -2j + 2k

4i -2j + 2k • (x-1)i + (y-2)j + (z-2)k = 0
2x-y+z = 2

For a fuller explanation, see

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