Determine whether the planes are parallel or orthogonal.
Equations given:
5x - 3y + z = 4
x + 4y + 7z = 1
How exactly do I solve this? It wasn't covered completely in class.
The normals of the two planes are
(5, -3, 1) and (1 , 4, 7)
Since one is not a multiple of the other, the two planes cannot be parallel
if they are orthogonal (perpendicular) then their dot product must be zero
so (5,-3,1)∙(1,4,7) = 5 -12 + 7 = 0
YEs, they are orthogonal
To determine whether the given planes are parallel or orthogonal, we need to compare the normal vectors of the planes.
Step 1: Write down the equations of both planes in the general form.
Plane 1: 5x - 3y + z = 4
Plane 2: x + 4y + 7z = 1
Step 2: Convert the equations to the vector form by expressing them as dot products with normal vectors.
Plane 1: (5, -3, 1) · (x, y, z) = 4
Plane 2: (1, 4, 7) · (x, y, z) = 1
Step 3: Compare the normal vectors of the planes.
The normal vector of Plane 1 is (5, -3, 1).
The normal vector of Plane 2 is (1, 4, 7).
Step 4: Determine if the planes are parallel or orthogonal based on the dot product of the normal vectors.
If the dot product is 0, the planes are orthogonal (perpendicular).
If the dot product is not 0, the planes are parallel.
Step 5: Calculate the dot product of the normal vectors.
(5, -3, 1) · (1, 4, 7) = (5)(1) + (-3)(4) + (1)(7) = 5 - 12 + 7 = 0
Step 6: Analyze the dot product's result.
Since the dot product of the normal vectors is 0, the planes are orthogonal.
Conclusion: The given planes are orthogonal.
To determine whether two planes are parallel or orthogonal, we need to consider their normal vectors.
Step 1: Rewrite the equations of the planes in the standard form, Ax + By + Cz = D, where the coefficients A, B, and C give the components of the normal vector to each plane.
For the first plane, 5x - 3y + z = 4, the coefficients are A = 5, B = -3, and C = 1.
For the second plane, x + 4y + 7z = 1, the coefficients are A = 1, B = 4, and C = 7.
Step 2: Identify the normal vectors of the planes by using the coefficients obtained in Step 1.
The normal vector of the first plane is [5, -3, 1].
The normal vector of the second plane is [1, 4, 7].
Step 3: Compare the normal vectors.
If the dot product of the normal vectors is zero, then the planes are orthogonal.
If the dot product of the normal vectors is nonzero, then the planes are not orthogonal. In this case, to determine if the planes are parallel or not, we can check if the ratios of the corresponding components of the normal vectors are equal. If the ratios are equal, the planes are parallel. Otherwise, they are not parallel.
Step 4: Calculate the dot product and ratios.
Dot product:
[5, -3, 1] · [1, 4, 7] = (5)(1) + (-3)(4) + (1)(7) = 5 - 12 + 7 = 0.
Ratios:
• The ratio of the x-components is 5/1 = 5.
• The ratio of the y-components is -3/4 = -3/4.
• The ratio of the z-components is 1/7.
Step 5: Analyze the results.
The dot product of the normal vectors is 0, which means the planes are orthogonal.
Since the ratios of the corresponding components of the normal vectors are not equal, the planes are not parallel.
Therefore, the planes are orthogonal but not parallel.