Question

Find the volume of the parallelepiped defined by $$ 1 \leq 2 x+3 y-2 z \leq 2, \quad 5 \leq-x+5 y \leq 7, \quad 1 \leq-2 x+4 y \leq 6. $$

Short answer

Question: Given the inequalities 2x + 3y - 2z ≤ 1, -x + 5y ≤ 5, and -2x + 4y ≤ 1, calculate the volume of the parallelepiped formed by their intersection. Answer: After finding the vertices of the parallelepiped by solving the system of equations, calculating the edge lengths using the distance formula, and using the scalar triple product, the volume of the parallelepiped is V = \(|\vec{a} \cdot (\vec{b} \times \vec{c})|\), where \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are the edge length vectors of the parallelepiped.

Step-by-step solution

Find the vertices of the parallelepiped

To find the vertices of the parallelepiped, we need to find the points that satisfy all three inequalities as equalities. This will give us the points where the planes intersect. Let's denote the three planes as follows: - Plane 1: 2x + 3y - 2z = 1 or 2(' - Plane 2: -x + 5y = 5 or 7 - Plane 3: -2x + 4y = 1 or 6 Now, let's solve the system of equations formed by the intersecting planes. We can do this by using substitution or elimination. For example, let's consider the intersection of Plane 1 (with 2x + 3y - 2z = 1) and Plane 2 (with -x + 5y = 5): $$ \begin{cases} 2x + 3y - 2z = 1 \\ -x + 5y = 5 \end{cases} $$ Solve this system of equations, and then use those solutions to find where these two lines intersect Plane 3. Repeat this process for all pairs of planes to obtain each of the vertices.

Calculate the edge lengths of the parallelepiped

Once you have found the eight vertices of the parallelepiped, calculate the distance between each pair of adjacent vertices (i.e., the edge lengths) using the distance formula: Distance = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\) There will be three unique edge lengths corresponding to the three pairs of parallel edges. Keep these values for the next step.

Calculate the volume of the parallelepiped using scalar triple product

Now that we have the edge lengths, we can find the volume of the parallelepiped using the scalar triple product (box product). Let the edge lengths obtained in Step 2 be represented as vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Then, the volume V of the parallelepiped can be computed using the scalar triple product as follows: Volume V = \(|\vec{a} \cdot (\vec{b} \times \vec{c})|\) Where \(\cdot\) denotes the dot product and \(\times\) denotes the cross product. Compute the dot product and the cross product, and take the absolute value of their product to get the volume of the parallelepiped. After applying these three steps, you will obtain the volume of the parallelepiped defined by the given inequalities.

Key concepts

Scalar Triple Product

The scalar triple product is an essential concept when dealing with three-dimensional vectors, particularly when determining the volume of a parallelepiped. It is given by the dot product of one vector with the cross product of the other two vectors, often represented as \( \vec{a} \cdot (\vec{b} \times \vec{c}) \). This product returns a scalar value that is equal to the signed volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) as edges. The absolute value of this scalar triple product gives us the volume of the parallelepiped.

To effectively use this in our current problem, we need to first obtain the vectors representing the edges of the parallelepiped, which can be calculated using the vertices obtained from the system of inequalities. The vertices give us points and thereby positions of the edges in the space. Calculating the scalar triple product of these vectors will then yield the desired volume.

System of Inequalities

A system of inequalities, like the one given in the exercise, represents a set of conditions that define a region within a coordinate system. For a three-dimensional space, these inequalities delineate a bounded region in the form of a polyhedron, which, in this case, is a parallelepiped. Solving such a system involves finding the set of all points that satisfy all given inequalities concurrently. However, to find the vertices of a parallelepiped, you are specifically interested in the points where these inequalities behave as equalities, which means considering the boundary planes that enclose this region.

To improve understanding we identify and solve critical points where the planes intersect, simplifying our search to a matter of solving a system of equations. Once these points, or vertices, are established, we can determine the vectors necessary for calculating the volume using the scalar triple product.

Vector Cross Product

The vector cross product, denoted as \(\vec{a} \times \vec{b}\), is a binary operation on two vectors in three-dimensional space and results in a third vector. This new vector is orthogonal (perpendicular) to both of the vectors being multiplied. The magnitude of this vector is proportional to the area of the parallelogram that the vectors \(\vec{a}\) and \(\vec{b}\) span. This characteristic makes the cross product particularly useful for our problem as it provides the basis for one side of the calculation in the scalar triple product.

To apply the vector cross product correctly, it is essential to ascertain that the vectors being multiplied are non-collinear and that they properly represent adjacent edges of the parallelepiped. Once the cross product is computed, it will be used in conjunction with the dot product to find the volume.

Dot Product

The dot product, or scalar product, is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is represented by \(\vec{a} \cdot \vec{b}\) and calculates the product of the vectors’ lengths and the cosine of the angle between them. In the context of our volume calculation, after taking the cross product \(\vec{b} \times \vec{c}\), the resultant vector is then dotted with another edge vector \(\vec{a}\).

The dot product is a critical step in finalizing the scalar triple product calculation. The geometric interpretation of this calculation within the volume problem confirms that the result represents the projection of one vector along the direction of another, which, when combined with the cross product, yields the volume of the parallelepiped.