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Magazine Chapter 11: Problem 20
Verify that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}=\) \((w \times u) \cdot v\) and find the volume of the parallelepiped (box) determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) (IMAGE CANNOT COPY) $$\mathbf{i}-\mathbf{j}+\mathbf{k} \quad 2 \mathbf{i}+\mathbf{j}-\mathbf{2 k} \quad-\mathbf{i}+2 \mathbf{j}-\mathbf{k}$$
Short Answer
Expert verified
The volume of the parallelepiped is 12.
Step by step solution
01
Identify the vectors
Given the vectors as follows:\( \mathbf{u} = \mathbf{i} - \mathbf{j} + \mathbf{k} \),\( \mathbf{v} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k} \),\( \mathbf{w} = -\mathbf{i} + 2\mathbf{j} - \mathbf{k} \).
02
Calculate \(\mathbf{u} \times \mathbf{v} \)
The cross product formula \( \mathbf{u} \times \mathbf{v} = \left| \begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \1 & -1 & 1 \2 & 1 & -2\end{array} \right| \) involves calculating the determinant:\[\mathbf{u} \times \mathbf{v} = (2 - (-1))\mathbf{i} - (1 + 2)\mathbf{j} + (1 + 2)\mathbf{k} = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}.\]
03
Compute \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\)
Using the result from Step 2, compute the dot product with \( \mathbf{w} \):\[(3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}) \cdot (-\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = (3)(-1) + (-3)(2) + (3)(-1) = -3 - 6 - 3 = -12.\]
04
Repeat cross and dot products for other permutations
Calculate \( \mathbf{v} \times \mathbf{w} \) and dot the result with \( \mathbf{u} \):\( \mathbf{v} \times \mathbf{w} = \left| \begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \2 & 1 & -2 \-1 & 2 & -1\end{array} \right| \)\( = (-1 - (-4))\mathbf{i} - (-2)(-2)\mathbf{j} + (4 - (-1))\mathbf{k} = 3\mathbf{i} - 4\mathbf{j} + 5\mathbf{k} \).Then, calculate:\[ (3\mathbf{i} - 4\mathbf{j} + 5\mathbf{k}) \cdot (\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3(1) + 4(1) + 5(1) = 3 - 4 + 5 = -12.\]
05
Verify cyclic nature
Check the cyclic permutation, using \( \mathbf{w} \times \mathbf{u} \) and dot it with \( \mathbf{v} \):The result of \( \mathbf{w} \times \mathbf{u} \) is:\[ \mathbf{w} \times \mathbf{u} = \left| \begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \-1 & 2 & -1 \1 & -1 & 1\end{array} \right| \]\( = (2 - (-1))\mathbf{i} - ((-1) + 1)\mathbf{j} + (1 + 2)\mathbf{k} = 3\mathbf{i} - 0\mathbf{j} + 3\mathbf{k} \).Dotting with \( \mathbf{v} \),\((3\mathbf{i} + 0\mathbf{j} + 3\mathbf{k}) \cdot (2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 6 + 0 - 6 = 0 + 0 - 6 = -6.\)
06
Calculate volume of parallelepiped
The volume is given by the absolute value of \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = -12\), hence the volume is \( | -12 | = 12 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross Product
The cross product, denoted by \( \mathbf{u} \times \mathbf{v} \), is an operation on two vectors in three-dimensional space. This operation results in a new vector that is orthogonal (or perpendicular) to the original two vectors. This is especially useful in physics and engineering for finding a vector perpendicular to a plane formed by two original vectors. The formula to compute the cross product involves determinants and the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \). For example:
- For vectors \( \mathbf{u} = a_1\mathbf{i} + b_1\mathbf{j} + c_1\mathbf{k} \) and \( \mathbf{v} = a_2\mathbf{i} + b_2\mathbf{j} + c_2\mathbf{k} \), the cross product is:\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \end{vmatrix} \]This results in a vector whose components are evaluated using the determinant method.
- The cross product is not commutative, meaning \( \mathbf{u} \times \mathbf{v} eq \mathbf{v} \times \mathbf{u} \). Instead, \( \mathbf{v} \times \mathbf{u} = - (\mathbf{u} \times \mathbf{v}) \).
Dot Product
The dot product, often represented as \( \mathbf{u} \cdot \mathbf{v} \), is a fundamental operation that yields a scalar. It involves multiplying corresponding components of two vectors and then summing the results. This is a measure of how much one vector extends in the direction of another.
- To compute the dot product for vectors \( \mathbf{u} = a_1\mathbf{i} + b_1\mathbf{j} + c_1\mathbf{k} \) and \( \mathbf{v} = a_2\mathbf{i} + b_2\mathbf{j} + c_2\mathbf{k} \), the formula is: \[ \mathbf{u} \cdot \mathbf{v} = a_1a_2 + b_1b_2 + c_1c_2 \]
- The dot product is commutative, so \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \).
- One important aspect of the dot product is that it results in a scalar, not a vector. If the dot product is zero, the vectors are orthogonal.
Parallelepiped Volume
The volume of a parallelepiped formed by three vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) is a geometric application of the scalar triple product. This is defined as \( (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} \). To find this volume:
- First, compute the cross product \( \mathbf{u} \times \mathbf{v} \) to get a vector perpendicular to the plane of \( \mathbf{u} \) and \( \mathbf{v} \).
- Then, take the dot product of this result with the third vector \( \mathbf{w} \), giving the volume of the parallelepiped.
- Mathematically, it's expressed as:\[ V = |(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}| \]
- The result is a scalar value representing the volume, and the absolute value ensures the volume is non-negative.
Determinants
A determinant is a special number that can be calculated from a matrix. It's widely used in linear algebra, especially for calculating cross products. For any 3x3 matrix formed by vectors \( \mathbf{u}, \mathbf{v} \), and \( \mathbf{w} \):
- The formula to compute a determinant for matrix\(\begin{vmatrix} a & b & c \ d & e & f \ g & h & i\end{vmatrix} \)is given by:\( a(ei-fh) - b(di-fg) + c(dh-eg) \)
- The determinant aids in calculating the cross product, where it forms the basis for finding orthogonal vectors.
- It also indicates the scaling factor or how volume changes when a transformation described by the matrix is applied.
