Determine whether the planes are parallel, perpendicular, or neither. them.(Round to one decimal place.) 9x-3y+6z=2, 2y=6x+4z
well, their normals have direction numbers
(9,-3,6) = (3,-1,2)
and (6,-2,4) = (3,-1,2)
so, what do you think?
parallel
To determine whether the planes are parallel, perpendicular, or neither, we can examine the coefficients of their normal vectors.
The equation of a plane can be written in the form: ax + by + cz = d, where (a, b, c) represents the normal vector of the plane.
For the first plane, 9x - 3y + 6z = 2, the coefficients of x, y, and z are 9, -3, and 6 respectively.
For the second plane, 2y = 6x + 4z, we need to rearrange the equation to match the standard form of the plane equation. Dividing both sides by 2, we get y = 3x + 2z. In this form, the coefficients of x, y, and z are 3, 1, and 2 respectively.
The normal vectors of the planes are [9, -3, 6] and [3, 1, 2] respectively.
For the planes to be parallel, their normal vectors must be scalar multiples of each other. In other words, if we multiply one normal vector by a scalar and get the other normal vector, the planes are parallel.
To check if two vectors are scalar multiples, we can compare their components. If the ratios of the corresponding components are the same, then the vectors are scalar multiples.
In this case, let's compare the ratios of the corresponding components:
For the x-component: 9/3 = 3
For the y-component: -3/1 = -3
For the z-component: 6/2 = 3
Since the ratios are not all the same, the planes are not parallel.
To determine if the planes are perpendicular, we can check if the dot product of their normal vectors is zero. If the dot product is zero, the planes are perpendicular.
The dot product of two vectors [a, b, c] and [d, e, f] is given by: ad + be + cf.
Calculating the dot product of [9, -3, 6] and [3, 1, 2], we get:
(9 * 3) + (-3 * 1) + (6 * 2) = 27 - 3 + 12 = 36
Since the dot product is not zero, the planes are not perpendicular either.
Therefore, based on the analysis, we can conclude that the planes represented by the equations 9x - 3y + 6z = 2 and 2y = 6x + 4z are neither parallel nor perpendicular.