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Chapter 9: Problem 1

Compute \(\mathbf{u} \times \mathbf{v}\) using the permutation symbol. Verify your answer by computing these products using traditional methods. a. \(\mathbf{u}=2 \mathbf{i}-3 \mathbf{k}, \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}\). b. \(\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{v}=\mathbf{i}-\mathbf{k}\). c. \(\mathbf{u}=5 \mathbf{i}+2 \mathbf{j}-3 \mathbf{k}, \mathbf{v}=\mathbf{i}-4 \mathbf{j}+2 \mathbf{k} .\)

Short Answer

Expert verified
The cross product for each pair of vectors using both methods should match, providing practice in both calculation methods and verification of the results.

Step by step solution

01

Compute Cross Product Using Permutation Symbol

The permutation symbol \(ε_{ijk}\) is a mathematical function used in the computation of cross products. It's equal to +1 if \(ijk\) is an even permutation of 123, -1 if it's an odd permutation, and 0 if any two indexes are the same. Use this formula: (\(\mathbf{u} \times \mathbf{v}\))_i = Σ_{j,k} ε_{ijk} u_j v_k . Do this for each part (a, b, c) in turn.
02

Verify Answer Using Traditional Methods

The traditional method for the cross product involves finding the determinant of a 3x3 matrix. Use this formula: \(\mathbf{u} \times \mathbf{v} = \mathbf{i}(u_2v_3 - u_3v_2) - \mathbf{j}(u_1v_3 - u_3v_1) + \mathbf{k}(u_1v_2 - u_2v_1)\). Insert the corresponding components of vectors \(\mathbf{u}\) and \(\mathbf{v}\) from each part (a, b, c) in the formula and calculate the determinant.

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Most popular questions from this chapter

Compute the following integrals: a. \(\int_{C}\left(x^{2}+y\right) d x+\left(3 x+y^{3}\right) d y\) for \(C\) the ellipse \(x^{2}+4 y^{2}=4\). b. \(\int_{S}(x-y) d y d z+\left(y^{2}+z^{2}\right) d z d x+\left(y-x^{2}\right) d x d y\) for \(S\) the positively oriented unit sphere. c. \(\int_{C}(y-z) d x+(3 x+z) d y+(x+2 y) d z\), where \(C\) is the curve of intersection between \(z=4-x^{2}-y^{2}\) and the plane \(x+y+z=0\). d. \(\int_{C} x^{2} y d x-x y^{2} d y\) for \(C\) a circle of radius 2 centered about the origin. e. \(\int_{S} x^{2} y d y d z+3 y^{2} d z d x-2 x z^{2} d x d y\), where \(S\) is the surface of the cube \([-1,1] \times[-1,1] \times[-1,1]\).

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