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Chapter 2: Problem 219

Consider the parallelepiped with edges \(O A, O B,\) and \( O C, \) where \( A(2,1,0), B(1,2,0),\) and \(C(0,1, \alpha).\) a. Find the real number \(\alpha>0\) such that the volume of the parallelepiped is 3 units \(^{3}.\) b. For \(\alpha=1,\) find the height \(h\) from vertex \(C\) of the parallelepiped. Sketch the parallelepiped.

Short Answer

Expert verified
(a) \(\alpha = 1\). (b) Height \(h = 1\).

Step by step solution

01

Determine the Volume Formula for a Parallelepiped

The volume of a parallelepiped with edges defined by vectors \( \vec{a}, \vec{b}, \vec{c} \) is given by the scalar triple product \( V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \). This represents the absolute value of the dot product between vector \( \vec{a} \) and the cross product of vectors \( \vec{b} \) and \( \vec{c} \).
02

Create Vectors from Points

Convert the points into vectors based from the origin: \( \vec{OA} = \langle 2, 1, 0 \rangle \), \( \vec{OB} = \langle 1, 2, 0 \rangle \), and \( \vec{OC} = \langle 0, 1, \alpha \rangle \). These vectors will be used to calculate the volume.
03

Calculate the Cross Product \((\vec{b} \times \vec{c})\)

Calculate the cross product \( \vec{b} \times \vec{c} \):\[(1, 2, 0) \times (0, 1, \alpha) = (2\alpha, -\alpha, 1)\]
04

Calculate the Dot Product

Next, find the dot product of \( \vec{a} \) and the cross product from Step 3:\[\vec{a} \cdot (\vec{b} \times \vec{c}) = (2, 1, 0) \cdot (2\alpha, -\alpha, 1) = 4\alpha - \alpha = 3\alpha\]
05

Solve for \(\alpha\) with Given Volume

Since the volume is provided as 3, set up the equation \[3\alpha = 3\]. Solving for \(\alpha\), divide both sides by 3 to obtain \(\alpha = 1\).
06

Find the Height from Vertex C with \(\alpha = 1\)

The height can be obtained using the formula \(h = \frac{V}{A_{base}}\), where \( A_{base} \) is the area of the parallelogram spanned by vectors \(\vec{a}\) and \(\vec{b}\). Calculate\( A_{base} = |\vec{a} \times \vec{b}| = |(2, 1, 0) \times (1, 2, 0)| = |(0, 0, 3)| = 3\). Then, \(h = \frac{3}{3} = 1\).
07

Sketch the Parallelepiped

To sketch the parallelepiped, draw a three-dimensional rectangular box and plot the vectors \(\vec{OA}\), \(\vec{OB}\), and \(\vec{OC}\) from one vertex, ensuring that \(\vec{OC}\) reaches the height determined in Step 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume of Parallelepiped
The volume of a parallelepiped is an important concept in three-dimensional geometry. It helps us measure the space enclosed within the figure, which has six faces that are parallelograms. This concept can be applied practically in calculating volumes in physics and engineering.

If you think of a parallelepiped, it's much like a more complex version of a box or a cube, where the sides slant rather than being perfectly perpendicular. To find its volume, we use a special operation involving vectors called the scalar triple product. This involves edges of the parallelepiped defined by vectors.
  • The formula for volume is given by the absolute value of the scalar triple product: \( V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \).
  • Here, \(\vec{a}, \vec{b}, \vec{c}\) are vectors representing the edges of the parallelepiped.
This formula essentially calculates how much area one base occupies in the direction of the third vector, considering the shape's slant through cross and dot products.
Cross Product
The cross product is a fundamental operation in vector algebra, particularly useful in physics and engineering. When dealing with vectors, the cross product allows us to find a vector that is perpendicular (or orthogonal) to two input vectors. This operation is essential in determining angles and areas in three-dimensional space.

The cross product of two vectors \(\vec{b}\) and \(\vec{c}\) is denoted as \(\vec{b} \times \vec{c}\). It results in a new vector that points in the direction perpendicular to both \(\vec{b}\) and \(\vec{c}\). This process involves a unique determinant calculation:
  • Use the right-hand rule to determine the direction of the resulting vector.
  • The magnitude of the cross product vector equals the area of the parallelogram spanned by the two vectors.
The calculation uses the components of the vectors to derive a new set of components that define this perpendicular vector. This operation is visually and mathematically invaluable when assessing the volume of a 3D shape like a parallelepiped.
Dot Product
The dot product is another essential vector operation, distinct from the cross product. It simplifies the process of finding the degree of alignment between two vectors. Essentially, it helps compute how much of one vector runs along another.When you take the dot product of two vectors, \(\vec{a}\) and \(\vec{b}\), the result is a scalar (a single number) rather than a vector, which measures the magnitude of projection of one vector onto the other:
  • The dot product of \(\vec{a}\) and \(\vec{b}\) is calculated as \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\).
  • This operation reflects the product of their magnitudes and the cosine of the angle between them.
In the context of a parallelepiped, the dot product combines with the cross product to find the scalar triple product, thus giving us the volume by determining how much the third vector contributes in the direction perpendicular to the parallelogram base.

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