please help turn this vector equation [x,y,z]=[3,7,-5]+s[1,2,-1]+t[1,-2,3] into a scalar equation
in the form of x+y+z+D=0
i understand in the case of [x,y]=[3,7,-5]+s[1,2,-1]
the direction vector is [1,2,-1] and the normal vector is [3,7,-5]
all i need to do in this case was to do
as the following
which would give me the scalar equation 3x+7y-5z-42=0
but how would you find the scalar equation if you have 2 direction vectors?
i have a quiz tommorwT_T but still dont get this question... OTL
hugs and kisses to anyone who can help/solve it for me....thanks in advance
A slightly different way: From the vector equation, x= 3+ s+ t, y= 7+ 2s- 2t, z= -5- s+ 3t. We can eliminate s by adding the first and third equations: x+ z= -2+ 4t. If we add two times the third equation to the second equation, we also eliminate s: y+ 2z= -3+ 4t. Finally, subtract y+ 2z=-3+ 4t from x+z= -2+ 4t to eliminate t: x+ z- y- 2z= x- y- z= 1. So the equation of the plane is x- y- z= 1, the same equation as before.