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Chapter 5: Problem 59

Prove that \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\)

Short Answer

Expert verified
The identity \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\) is proven by applying the definition of cross product and dot product.

Step by step solution

01

Write vectors in Component Form

Let \(\mathbf{u} = (u1,u2,u3)\), \(\mathbf{v} = (v1,v2,v3)\), and \(\mathbf{w} = (w1,w2,w3)\) be in three-dimensional Cartesian coordinates.
02

Evaluate the Cross Product

The cross product of two vectors \(\mathbf{v} = (v1,v2,v3)\) and \(\mathbf{w} = (w1,w2,w3)\) is given by \( \mathbf{v} \times \mathbf{w} = (v2w3 - v3w2, v3w1 - v1w3, v1w2 - v2w1)\).
03

Compute \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\)

The dot product is computed as \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) = u1(v2w3 - v3w2)+u2(v3w1 - v1w3)+u3(v1w2 - v2w1)\).
04

Evaluate the Cross Product for \(\mathbf{u} \times \mathbf{v}\)

The cross product of \(\mathbf{u} = (u1,u2,u3)\) and \(\mathbf{v} = (v1,v2,v3)\) is \(\mathbf{u} \times \mathbf{v} = (u2v3-u3v2, u3v1-u1v3, u1v2-u2v1)\). It represents a new vector.
05

Compute \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\)

The dot product is given by \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}= w1(u2v3-u3v2)+w2(u3v1-u1v3)+w3(u1v2-u2v1)\).
06

Conclusion

From the calculations in Step 3 and Step 5, It can be seen that the result of \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\) is the same, which proves the identity.

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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 is an operation on two vectors in three-dimensional space. It results in a third vector that is perpendicular to the plane formed by the two input vectors. The cross product is denoted as \( \mathbf{v} \times \mathbf{w} \), where \( \mathbf{v} \) and \( \mathbf{w} \) are vectors.
  • The direction of the resulting vector is determined by the right-hand rule. If you point your index finger along \( \mathbf{v} \) and your middle finger along \( \mathbf{w} \), your thumb points in the direction of the cross product.
  • The magnitude of the cross product vector is equal to the area of the parallelogram with sides \( \mathbf{v} \) and \( \mathbf{w} \).
When computing the cross product in component form, the formula is:\[\mathbf{v} \times \mathbf{w} = (v_2 w_3 - v_3 w_2, v_3 w_1 - v_1 w_3, v_1 w_2 - v_2 w_1) \]This vector operation is particularly useful in physics and engineering, where directions and magnitudes are key.
Dot Product
The dot product is another fundamental vector operation used to compute the product of two vectors. Unlike the cross product, the dot product results in a scalar value rather than a vector.
  • It measures the extent to which two vectors are parallel.
  • Mathematically expressed as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
The dot product is key in various applications, such as calculating angles and projections. It is essential for verifying vector identities, like the one given in the exercise. This operation is often called the scalar product because the result is a single number.
Cartesian Coordinates
Cartesian coordinates describe the position of a point or object in three-dimensional space using three values, usually labeled \( (x, y, z) \). Each value corresponds to the position along one of the three axes in 3D space.
  • With vectors represented as \( (u_1, u_2, u_3) \), calculations become straightforward as each element corresponds directly to one spatial dimension.
  • The system uses perpendicular axes that intersect at a common origin. Each coordinate measures distance from this origin along a specific axis.
Using Cartesian coordinates simplifies vector operations like cross and dot products. They provide a clear, universal framework for representing vectors and calculating their interactions.
Vector Identity
In vector algebra, identities are equations that hold true for all vectors involved in specific vector operations. The triple product identity given in the exercise is a common and crucial one:\[\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\]
  • Both sides of the equation represent the scalar triple product, essentially expressing the volume of the parallelepiped determined by the three vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \).
  • This identity illustrates how the order of operations can change without affecting the outcome when combining cross and dot products.
Understanding vector identities is crucial for solving problems involving complex vector operations, as they often simplify the calculations required.

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Most popular questions from this chapter

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The length or norm of a vector is \(\|\mathbf{v}\|=\left|v_{1}+v_{2}+v_{3}+\cdots+v_{n}\right|\) (b) The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is another vector represented by \(\mathbf{u} \cdot \mathbf{v}=\left(u_{1} v_{1}, u_{2} v_{2}, u_{3} v_{3}, \ldots, u_{n} v_{n}\right)\)

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