Problem 33 - APEX Calculus

Question

In Exercises 33-34, find the volume of the parallelepiped defined by the given vectors. \(\vec{u}=\langle 1,1,1\rangle, \quad \vec{v}=\langle 1,2,3\rangle, \quad \vec{w}=\langle 1,0,1\rangle\)

Step-by-Step Solution

Answer

The volume of the parallelepiped is 2.

Step 1: Calculate Cross Product

To find the volume of a parallelepiped defined by vectors \(\vec{u}, \vec{v},\) and \(\vec{w}\), we first compute the cross product of \(\vec{v}\) and \(\vec{w}\). The cross product formula for two vectors \(\vec{a} = \langle a_1, a_2, a_3 \rangle\) and \(\vec{b} = \langle b_1, b_2, b_3 \rangle\) is given by: \[\vec{a} \times \vec{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle\]Substituting \(\vec{v} = \langle 1,2,3 \rangle\) and \(\vec{w} = \langle 1,0,1 \rangle\), we calculate:\[\vec{v} \times \vec{w} = \langle 2 \cdot 1 - 3 \cdot 0, 3 \cdot 1 - 1 \cdot 1, 1 \cdot 0 - 2 \cdot 1 \rangle = \langle 2, 2, -2 \rangle\]

Step 2: Compute Dot Product

With the cross product \(\vec{v} \times \vec{w} = \langle 2, 2, -2 \rangle\) computed, proceed to find the dot product of this result with \(\vec{u} = \langle 1, 1, 1 \rangle\). The dot product of vectors \(\vec{a} = \langle a_1, a_2, a_3 \rangle\) and \(\vec{b} = \langle b_1, b_2, b_3 \rangle\) is:\[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\]Therefore, we have:\[\vec{u} \cdot (\vec{v} \times \vec{w}) = 1\cdot2 + 1\cdot2 + 1\cdot(-2) = 2 + 2 - 2 = 2\]

Step 3: Find Volume

The volume of the parallelepiped is the absolute value of the scalar triple product, which is the result of the dot product calculated earlier. Thus, the volume is:\[|\vec{u} \cdot (\vec{v} \times \vec{w})| = |2| = 2\]

Key Concepts

Vector Cross Product

In the world of 3D vectors, the cross product is a powerful tool for finding a vector that is perpendicular to two given vectors. This is especially useful in fields such as physics and engineering, where calculating perpendicular directions is often necessary. For vectors \( \vec{a} = \langle a_1, a_2, a_3 \rangle \) and \( \vec{b} = \langle b_1, b_2, b_3 \rangle \), the cross product \( \vec{a} \times \vec{b} \) is defined as:\[\vec{a} \times \vec{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \]

• This operation is unique because the resulting vector is orthogonal to the original two vectors.
• The magnitude of this vector gives you the area of the parallelogram formed by \( \vec{a} \) and \( \vec{b} \).

When calculating the volume of a parallelepiped, this cross product provides a pivotal step by simplifying complex 3D interactions.

Scalar Triple Product

The scalar triple product is a specific operation used to calculate the volume of a parallelepiped formed by three vectors. It combines both the cross product and the dot product. If you have vectors \( \vec{u}, \vec{v}, \) and \( \vec{w} \), the scalar triple product is determined as:\[ \vec{u} \cdot (\vec{v} \times \vec{w}) \]

• First, find the cross product of two vectors to get a new vector.
• Then, compute the dot product of this new vector with the third vector.
• The result is a scalar value, hence the name "scalar triple product".
• The absolute value of this scalar gives you the volume of the parallelepiped.

This sequence serves as the structural backbone for finding volumes in 3D vector spaces.

Dot Product

The dot product is a fundamental operation in vector algebra, used to find the angle and other properties between two vectors. For vectors \( \vec{a} = \langle a_1, a_2, a_3 \rangle \) and \( \vec{b} = \langle b_1, b_2, b_3 \rangle \), the dot product \( \vec{a} \cdot \vec{b} \) is given by:\[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

• This product results in a scalar, unlike the cross product, which results in a vector.
• The dot product helps determine how "aligned" two vectors are to each other.
• A dot product of zero implies that the vectors are orthogonal (perpendicular).

In the context of calculating volumes, the dot product is an integral step of the scalar triple product, making it crucial for determining the volume of a parallelepiped.

3D Vectors

Vectors in three-dimensional space, often referred to as 3D vectors, are essential in representing physical quantities like force, velocity, and position. A 3D vector \( \vec{v} \) can be expressed as \( \langle x, y, z \rangle \), containing three components:

• x-component: Represents the vector's horizontal distance.
• y-component: Shows its vertical distance.
• z-component: Indicates "depth," or the distance into/out of the plane.

3D vectors are incredibly useful in physics for modeling real-life problems where interactions occur in three-dimensional space. By using operations like cross products and dot products, information about magnitude, direction, and orientation can be extracted. This makes 3D vectors invaluable tools in various scientific and engineering disciplines, providing a framework for calculating properties like the volume of a parallelepiped.