Problem 21
Question
Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2},\) and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ 5 x-3 y+z=4, x+4 y+7 z=1 $$
Step-by-Step Solution
Verified Answer
The planes \(5x - 3y + z = 4\) and \(x + 4y + 7z = 1\) are perpendicular.
1Step 1: Analyze the coefficients of the planes
Identify the coefficients of the two planes. For the first plane \(5x - 3y + z = 4\), the coefficients \(a_1, b_1, c_1\) are 5, -3, and 1 respectively. For the second plane \(x + 4y + 7z = 1\), the coefficients \(a_2, b_2, c_2\) are 1, 4, and 7 respectively.
2Step 2: Test for parallelism
The planes are parallel if \(a_1 = k a_2 , b_1 = k b_2 , c_1 = k c_2\) for some constant \(k\). Divide \(a_1\) by \(a_2\), \(b_1\) by \(b_2\), and \(c_1\) by \(c_2\). If the results are equal, then the planes are parallel. In this case, dividing 5 by 1, -3 by 4, and 1 by 7 gives 5, -0.75 and approximately 0.14. These are not equal, therefore the planes are not parallel.
3Step 3: Test for perpendicularity
The planes are perpendicular if \(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\). In this case, \(5*1 - 3*4 + 1*7 = 5 - 12 + 7 = 0\). Therefore, the planes are perpendicular.
Key Concepts
Parallel PlanesEquation of a PlaneVector Algebra
Parallel Planes
Parallel planes occur when two planes extend in the same direction without ever meeting, very much like parallel lines. For two planes to be parallel, each coefficient of the plane in the general form equation must be proportional to the corresponding coefficient in the other plane. To think about it more simplistically, two planes are parallel if they have identical normal vectors, scaled by a constant. The normal vector of a plane can be deduced from the coefficients of the equation in the form \(ax + by + cz = d\).
This idea is crucial because the normal vector essentially determines the entire orientation of the plane. Imagine you have two planes expressed by equations in your exercise: \(5x - 3y + z = 4\) and \(x + 4y + 7z = 1\). Here, the task is to find a constant \(k\) such that each coefficient of the first equation equals \(k\) times the corresponding coefficient in the second equation.
We compute this by dividing the coefficients:
This idea is crucial because the normal vector essentially determines the entire orientation of the plane. Imagine you have two planes expressed by equations in your exercise: \(5x - 3y + z = 4\) and \(x + 4y + 7z = 1\). Here, the task is to find a constant \(k\) such that each coefficient of the first equation equals \(k\) times the corresponding coefficient in the second equation.
We compute this by dividing the coefficients:
- For \(a_1/a_2 = 5/1 = 5\)
- For \(b_1/b_2 = -3/4\)
- For \(c_1/c_2 = 1/7\)
Equation of a Plane
The equation of a plane is a formula that captures the set of all points extending through three-dimensional space. It typically appears in the form \(ax + by + cz = d\), which looks simple but contains rich geometric information.
The coefficients \(a\), \(b\), and \(c\) reveal the normal vector to the plane. This vector is crucial because it determines the direction perpendicular to the plane itself. The variable \(d\) represents the distance from the origin to the plane along this normal vector, adjusting the plane's position in space.
Understanding this formula helps us unpack complex spatial relationships effectively. Taking a plane like \(5x - 3y + z = 4\), we can identify the normal vector as \((5, -3, 1)\). This enables us to calculate aspects like angles, distances, or even intersections with other planes or lines.
The coefficients \(a\), \(b\), and \(c\) reveal the normal vector to the plane. This vector is crucial because it determines the direction perpendicular to the plane itself. The variable \(d\) represents the distance from the origin to the plane along this normal vector, adjusting the plane's position in space.
Understanding this formula helps us unpack complex spatial relationships effectively. Taking a plane like \(5x - 3y + z = 4\), we can identify the normal vector as \((5, -3, 1)\). This enables us to calculate aspects like angles, distances, or even intersections with other planes or lines.
Vector Algebra
Vector algebra is a branch of mathematics focusing on vectors, which are powerful tools used to model quantities possessing both magnitude and direction. Vectors are pivotal in understanding plane equations, as each plane's orientation is dictated by its normal vector.
In the realm of planes, vector algebra aids in several analyses, such as checking for parallelism or perpendicularity between planes. To test perpendicularity, for instance, the dot product of their normal vectors should be zero. This understanding stems from the fact that a zero dot product signifies orthogonality. For planes \(5x - 3y + z = 4\) and \(x + 4y + 7z = 1\), their normal vectors are \((5, -3, 1)\) and \((1, 4, 7)\) respectively.
The dot product is calculated as:
In the realm of planes, vector algebra aids in several analyses, such as checking for parallelism or perpendicularity between planes. To test perpendicularity, for instance, the dot product of their normal vectors should be zero. This understanding stems from the fact that a zero dot product signifies orthogonality. For planes \(5x - 3y + z = 4\) and \(x + 4y + 7z = 1\), their normal vectors are \((5, -3, 1)\) and \((1, 4, 7)\) respectively.
The dot product is calculated as:
- \(5 \times 1 + (-3) \times 4 + 1 \times 7 = 5 - 12 + 7 = 0\)
Other exercises in this chapter
Problem 21
Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ f(x, y)=x^{2}+y^{2} $$
View solution Problem 21
Use Lagrange multipliers to find the given extremum of \(f\) subject to two constraints. In each case, assume that \(x, y,\) and \(z\) are nonnegative. $$ \begi
View solution Problem 22
Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function \(f(x, y)
View solution Problem 22
Evaluate the double integral. $$ \int_{-1}^{1} \int_{-2}^{2}\left(x^{2}-y^{2}\right) d y d x $$
View solution