Question

find the points where the straight line joining the points (2,-3,1) and (3,-4,5) cuts the plane 3x+y+z=8.

Answer 1

we need to know the vector between the points

Answer 2

is it going to be
\[\frac{ x-2 }{ 1 }=\frac{ y+3 }{ -1 }=\frac{ z-1 }{ 4 }\] ????

Answer 3

possibly, let me try my way to verify; just subtract one point from the other

(3,-4,5)
- (2,-3,1)
---------
1,-1,4 , nice

so lets form the component equations with this vector and one of the given points

Answer 4

which point is your most favorite :)

Answer 5

let's go with (3,-4,5) :-)

Answer 6

ok, using the point and a scalar to stretch the vector we get:

x = 3 + 1t
y = -4 - 1t
z = 5 + 4t

notice how we used the point parts and the vector parts to create equation that we can use to find any point on the line; also we can substitute these equations into the xyz parts of the plane to solve for "t"

Answer 7

3x + y + z = 8

3(3+t) + (-4-t) + (5+4t) = 8

now what is t?

Answer 8

so if we get the value of t we put it in the x y z equations and we get the point of intersection!! Nice!! t is coming=-1/3.

Answer 9

good, then solve for xyz :)

Answer 10

my answers are coming
x=8/3
y=-11/3
z=19/3
But the answer given in my book is(1,-2,7).

Answer 11

z = 11/3

http://www.wolframalpha.com/input/?i=x+%3D++3+%2B+t%2C+y+%3D+-4+-+t%2C+z+%3D++5++%2B+4t+%2C+t%3D-1%2F3

with the information you have presented in your post, this is the correct answer; you can double check by inserting the found xyz values into the plane equation

make sure you havent typed an error, and also that you are looking at the correct solution key

Answer 12

sorry!! z is coming 11/3!

Answer 13

i think you are right!! the solution's incorrect !! anyways a big thanks for guiding me in such a systematic way!! Cheers to u!! :-)

Answer 14

youre welcome :) good luck