Math  Equations of plane(check)
posted by Shaila .
Consider the plane that contains points A(2,3,1), B(11,1,2), C(7,3,6).
a) Find two vectors that are parallel to the plane.
Ans: AC, BC or AB will be parallel to the plane.
b) Find two vectors that are perpendicular to the plane.
Ans: if i find the normal vector then i can use dot product and use arbitrary values to find the two vectors
c) Write a vector equation of the plane
Ans: I have two points AC, BC or AB and i have the normal, so i use them to write the vector equation.
d) Write the scalar equation of the plane.
Ans: the normal vectors are my coefficients for the(x,y,z) therefore if i substitute a point to find the constant 'd'
e) Write an equation of the line through the x and y intercepts of the plane
Ans: Find the x and y intercepts and those will be my points my z value being zero and using the normal as direction vector
Are the answers correct? I am a little confused, please correct if i am wrong somewhere.

I actually did this problem
a) yes
b) the simplest normal is [1,5,3) so any other multiple of that would do
c) [x,y,z] = [2,3,1] + s[13,2,1] + t[9,6,7]
d) I got x  5y + 3z = 10 , all 3 original points satisfied
e) for xintercept, let both y and z = 0, so x=10
for the point (10,0,0)
for yintercept let both x and z = 0, y = 2
for the point (0,2,0)
direction vector of line joining the intercepts is [10, 2,0] or [5,1,0]
so equation of line is [x,y,z] = 0,2,0) + t[5,1,0]
check my arithmetic, last time I made a silly error if you recall 
for the e part of the question you mean:
[x,y,z]=[2,3,1]+t[10,2,0]+s[5,1,0]
right??
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