Multivariable Calculus: Vectors and the Geometry of Space
Find parametric equations for the line through (5,1,0) that is perpendicular to the plane 2x-y+z=1. Also, in what points does this line intersect the coordinate planes.
Please explain the steps. (:
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Yes, for a parametric equation of a line the equation of line is actually a combination of 3 equations
x = 1 + ar
y = 1 + br
z = 1 + cr
Now, this line passes through the point (5,1,0) hence putting x = 5, y = 1 and z = 0
5 = 1 + ar => a = 4 / r
1 = 1 + br => br = 0 => b = 0 (as r not equal to zero from other two equations)
0 = 1 + cr => c = -1 / r
Now for a line to e perpendicular to a plane, the line vector ((a,b,c) in this case) and the normal vector of the line (2, -1, 1) are parallel.
Which means their dot product has a value 1 (since cos0 = 1)
hence, having the dot product = 0 we have
a.2 + b. (-1) + c.1 = 1
Replacing a by 4 / r and c by -1 / r and b = 0. We have
8 / r - 1 / r = 1 => r = 7
Hence, a = 4 / 7 and c = -1 / 7
Hence, the required equation of the line is
x = 1 + 4r / 7
y = 1
z = 1 - r / 7
For crossing XY planes, Z = 0 hence r = 7, x = 5, y = 1 point (5,1,0) [our point]
For Crossing XZ planes, Y = 0, this condition is never met, the line does not cross XZ plane
For crossing YZ plane, X = 0 hence 1 + 4r / 7 = 0, r = - 7 / 4, hence x = 0, y = 1, z = 1.25 point is (0,1,1.25)
Thank you so much! I just have one more question. How did you figure out that x=1+ar, y=1+br, and z=1+zr. I'm wondering where you got the 1 on all of them. Thanks. (:
The 1 is unitary displacement in each direction, since the locus of the point moves linearly in all 3 directions hence, that is handled by adding 1. We could use any constant in place of 1 here, but the values of r will change in that case, but the final answer will be in the multiples of the a,b and c found out.
Please let me know if you need more explanation.
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There are two three corrections, which are typo in the original solutions -
"Now for a line to e perpendicular to a plane, the line vector ((a,b,c) in this case) and the normal vector of the line (Should read Plane) (2, -1, 1) are parallel."
"hence, having the dot product = 0 (should read 1) we have"
Oh ok I see. So in place of the 1 i could use any number? And that would change the r, but in the end, the answer would still be correct?
Hey I am sorry, let me check it a little more. I will come back to you.
However, we can use 1, that is the general convention