Explain why the planes 3x-6y-9z+5=0 and x+2=2y+3z never intersect
3x-6y-9z = -5
x-2y-3z = -2
are clearly parallel planes.
To determine whether the given planes intersect, we can compare their normal vectors.
The given equation of the first plane, 3x - 6y - 9z + 5 = 0, can be rewritten as 3x - 6y - 9z = -5.
From this equation, we can identify the normal vector of the first plane as ⟨3, -6, -9⟩.
The normal vector of the second plane x + 2 = 2y + 3z is ⟨1, -2, -3⟩.
For the planes to intersect, their normal vectors would need to be linearly dependent, meaning they must be scalar multiples of each other.
However, the normal vectors ⟨3, -6, -9⟩ and ⟨1, -2, -3⟩ are not scalar multiples of each other.
Therefore, the two planes never intersect.
To understand why the planes 3x-6y-9z+5=0 and x+2=2y+3z never intersect, we can compare the equations of the two planes.
The equation of the first plane is 3x-6y-9z+5=0, while the equation of the second plane can be rearranged as x-2y-3z=2.
We know that two planes are parallel if their normal vectors are parallel. The normal vectors of the two planes are the coefficients of x, y, and z in their respective equations.
For the first plane, the normal vector is [3, -6, -9] because it is the coefficients of x, y, and z in the equation 3x-6y-9z+5=0.
For the second plane, the normal vector is [1, -2, -3] because it is the coefficients of x, y, and z in the equation x-2y-3z=2.
Now we can compare the normal vectors [3, -6, -9] and [1, -2, -3].
If two vectors are parallel, one vector is a scalar multiple of the other. In other words, if [3, -6, -9] is a scalar multiple of [1, -2, -3], then the planes are parallel.
To check if they are scalar multiples, we calculate the ratio of the corresponding components of the two vectors:
3/1 = -6/-2 = -9/-3.
We find that all the ratios are equal (-3), which means the two normal vectors are scalar multiples of each other.
Since the planes have parallel normal vectors, it implies that they are parallel to each other. Parallel planes do not intersect, so the planes 3x-6y-9z+5=0 and x+2=2y+3z never intersect.