through the points P(1, 2, 1) & Q(2, 1, 4), and is parallel to the x-axis.
I managed to get the right answer using the unit vector for the x-axis [1, 0, 0], the vector PQ and one of those two points, but wouldn't any vector parallel to the x-axis work? Isn't [5, 0, 0] also parallel to the x-axis? Multiplying the unit vector by any scalar value would give an appropriate vector, no?
Does this mean that there are more than one equation for the same plane? or is it that there are several planes that satisfy those conditions (through those two points and parallel to x-axis)?
I managed to get the right answer using the unit vector for the x-axis [1, 0, 0], the vector PQ and one of those two points, but wouldn't any vector parallel to the x-axis work? Isn't [5, 0, 0] also parallel to the x-axis? Multiplying the unit vector by any scalar value would give an appropriate vector, no?
Does this mean that there are more than one equation for the same plane? or is it that there are several planes that satisfy those conditions (through those two points and parallel to x-axis)?
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there are infinite number of equations to a plane;
for example; x-2y+z=0 same as 5x-10y+5z=0 same as 1.5x-3y+1.5z=0 you can multiply the equation with any parameter belonging to real numbers;
notice that the coefficients of x, y , and z are coordinates to a vector perpendicular to the plane;
and there are infinite numbers of vectors that are perpendicular to the plane
for example; x-2y+z=0 same as 5x-10y+5z=0 same as 1.5x-3y+1.5z=0 you can multiply the equation with any parameter belonging to real numbers;
notice that the coefficients of x, y , and z are coordinates to a vector perpendicular to the plane;
and there are infinite numbers of vectors that are perpendicular to the plane
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a plane is given by ax + by +cz = k where N =[a,b,c] is normal (vector) of plane
so is 2ax + 2by +2cz = 2k is same plane in fact we can use any non-trivial multiplier.
a vector [5,0,0] ~ [1,0,0] i.e. has same direction and can be used to find other directions as you have calculated when working in vector geometry.
Although we try to simplify use [1,0,0].
Watch out for questions that require a metric (distance formulae etc.)
so is 2ax + 2by +2cz = 2k is same plane in fact we can use any non-trivial multiplier.
a vector [5,0,0] ~ [1,0,0] i.e. has same direction and can be used to find other directions as you have calculated when working in vector geometry.
Although we try to simplify use [1,0,0].
Watch out for questions that require a metric (distance formulae etc.)