) Determine whether the following two planes x + 4y − z = 7 and 5x − 3y −7z = 11 are parallel, orthogonal, coincident (that is, the same) or none of these.
Their normals are
(1,4,-1) and (5,-3,-7) respectively
clearly they are not parallel, or one of the normals would be a multiple of the other
nor are they coincident
let's look at their dot product:
(1,4,-1) . (5,-3,-7)
= 5 - 12 + 7 = 0
So they are perpendicular or orthogonal.
To determine the relationship between two planes, we need to examine the coefficients of the x, y, and z variables in the equations of the planes.
Let's first write the equations of the given planes in the general form Ax + By + Cz + D = 0:
Plane 1: x + 4y − z − 7 = 0 (Equation 1)
Plane 2: 5x − 3y − 7z − 11 = 0 (Equation 2)
Now we will compare the normal vectors of both planes, which are the coefficients of x, y, and z in the equations. The normal vector of Plane 1 is (1, 4, -1), and the normal vector of Plane 2 is (5, -3, -7).
Parallel planes have the same normal vector or proportional normal vectors. Orthogonal planes have normal vectors that are perpendicular (dot product is zero).
To determine if the planes are parallel, we need to compare their normal vectors. If the normal vectors are proportional, then the planes are parallel. We can determine proportionality by comparing the ratios of the corresponding components. In this case, we check if the ratios of (1/5), (4/-3), and (-1/-7) are equal:
1/5 = 4/-3 = -1/-7
Simplifying the ratios, we have:
1/5 = -4/3 = 1/7
Since these ratios are not equal, the planes are not parallel.
Next, we'll check if the planes are orthogonal (perpendicular). To determine if the normal vectors are perpendicular, we need to calculate their dot product. If the dot product is zero, the planes are orthogonal.
Calculating the dot product of the normal vectors:
(1)(5) + (4)(-3) + (-1)(-7) = 5 - 12 + 7 = 0
Since the dot product is zero, the planes are orthogonal.
Therefore, the given planes x + 4y − z = 7 and 5x − 3y − 7z = 11 are orthogonal.