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 $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$. $$x-5y-z=1,5x-25y-5z=-3$$

Short answer

The two given planes are parallel and not perpendicular.

Step-by-step solution

Identification of Coefficients

Identify the coefficients for each term in the plane equations. For equation 1 which is $x-5y-z=1$, the coefficients $a_1,b_1,c_1$ are 1, -5, and -1. For equation 2 which is $5x-25y-5z=-3$, the coefficients $a_2,b_2,c_2$ are 5, -25, and -5.

Test for Parallelism

Test to see if the planes are parallel by checking if the coefficients from equation 1 are in a constant ratio with the coefficients from equation 2. This would mean that $a_1/a_2=b_1/b_2=c_1/c_2$. Here 1/5 equals to -5/-25 and equals to -1/-5, that is 0.2, hence the planes are indeed parallel.

Test for Perpendicularity

Test for perpendicularity by using the formula $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$. If the result is zero, the planes are perpendicular. Here the result is $1\ast 5+(-5)\ast (-25)+(-1)\ast -5=5+125+5=135$, which is not equal to 0, hence the planes are not perpendicular.

Key concepts

Parallel and Perpendicular Planes

Understanding the orientation of planes in three-dimensional space is crucial for various applications in mathematics and physics. Two planes can either be parallel, perpendicular, or neither. For planes described by equations in the form of $ax+by+cz=d$, we determine their orientation by analyzing their normal vectors.

The normal vector of a plane is a vector that is perpendicular to the surface of the plane. It is denoted through the coefficients $a,b,c$ in its equation. For example, in our exercise, the normal vectors for the given planes are $1,-5,-1$ and $5,-25,-5$, respectively.

Two planes are parallel if their normal vectors are scalar multiples of each other, which means one vector can be obtained by multiplying the other by a nonzero constant. If the direction ratios of the normal vectors are proportional, i.e., $a_1/a_2=b_1/b_2=c_1/c_2$, then the planes are parallel. In the given exercise, we observe that the ratio $0.2$ is consistent across all coefficient pairs, indicating parallelism.

For perpendicular planes, their normal vectors are orthogonal to each other. This is established when the dot product of the normal vectors is zero, $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$. However, in our example, the dot product is 135, which shows that the planes are not perpendicular.

Plane Coefficients

The coefficients of a plane's equation $ax+by+cz=d$ serve as a defining feature for the plane's properties, determining not only its orientation in space but also its distance from the origin and its inclination.

In our exercise, the coefficients of the first plane are $a_1=1,b_1=-5,c_1=-1$, and for the second plane $a_2=5,b_2=-25,c_2=-5$. These coefficients represent the scalar components of the plane's normal vector, thus dictating the direction in which the plane ‘stands’.

It is also important to note that the distance of a plane from the origin can be determined by the coefficients and the constant term in its equation, through the formula $d=\frac{|a{x}_0+b{y}_0+c{z}_0-d|}{\sqrt{a^2+b^2+c^2}}$, where $x_0,y_0,z_0$ is the origin. This is a valuable piece of information for analyzing geometric relationships between planes and other figures in space.

Vector Algebra

Vector algebra is an area of mathematics that deals with quantities that have both magnitude and direction. It is a fundamental concept useful in studying the geometry of shapes in multiple dimensions, including the features of planes and their orientations relating to one another.

When dealing with planes, we frequently use vectors to represent normal vectors as mentioned above. Vectors are also essential in operations such as the dot product, which helps in determining the angle between two planes. To further examine this concept, consider the dot product operation carried out in our exercise (\texttt{Step 3}): $1\ast 5+(-5)\ast (-25)+(-1)\ast -5=135$, which signifies that the two planes are not at a right angle to each other because the dot product is not zero.

Additionally, vector algebra involves cross product, vector addition, scalar multiplication, and other operations that are useful in the broader context of 3D space geometry. In physics, vector algebra is extensively used to understand forces, velocity, and other vector quantities that influence the natural world.