Problem 27 - Thomas Calculus

Question

Which of the following are always true, and which are not always true? Give reasons for your answers. a. \(|\mathbf{u}|=\sqrt{\mathbf{u} \cdot \mathbf{u}} \quad\) b. \(\mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|\) c. \(\mathbf{u} \times \mathbf{0}=\mathbf{0} \times \mathbf{u}=\mathbf{0} \quad\) d. \(\mathbf{u} \times(-\mathbf{u})=\mathbf{0}\) e. \(\mathbf{u} \times \mathbf{v}=\mathbf{v} \times \mathbf{u}\) f. \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=\mathbf{u} \times \mathbf{v}+\mathbf{u} \times \mathbf{w}\) g. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{v}=0\) h. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\)

Step-by-Step Solution

Answer

a, c, d, f, g, h are true. b, e are not true.

Step 1: Evaluate statement a

We need to determine whether \(|\mathbf{u}|=\sqrt{\mathbf{u} \cdot \mathbf{u}}\). According to the properties of vectors, the magnitude of vector \(\mathbf{u}\) is given by this formula. Hence, this statement is always true.

Step 2: Assess statement b

Let's see if \(\mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|\). The left-hand side, \(\mathbf{u} \cdot \mathbf{u}\), equals \(|\mathbf{u}|^2\), not \(|\mathbf{u}|\). Therefore, this statement is not always true.

Step 3: Evaluate statement c

We need to see if \(\mathbf{u} \times \mathbf{0}=\mathbf{0} \times \mathbf{u}=\mathbf{0}\). This is true because any vector crossed with the zero vector results in the zero vector. So, this statement is always true.

Step 4: Check statement d

Is \(\mathbf{u} \times(-\mathbf{u})=\mathbf{0}\)? The cross product of any vector with itself or its negative is the zero vector. This statement is always true.

Step 5: Verify statement e

For \(\mathbf{u} \times \mathbf{v} = \mathbf{v} \times \mathbf{u}\), consider the anti-commutative property of the cross product. This states \(\mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u})\). Therefore, this statement is not always true.

Step 6: Assess statement f

Check if \(\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}\). The distributive property of the cross product over vector addition dictates this identity is always true.

Step 7: Evaluate statement g

For \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{v} = 0\), note the cross product \(\mathbf{u} \times \mathbf{v}\) is perpendicular to \(\mathbf{v}\), meaning their dot product is zero. Hence, this statement is always true.

Step 8: Verify statement h

Check if \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = \mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\). This is known as the vector triple product expansion (or Lagrange's identity), and is always true.

Key Concepts

Vector Properties

Vectors are fundamental in physics and engineering, representing quantities having both magnitude and direction. They can be represented graphically as arrows pointing from one point to another in a coordinate system.

Key properties of vectors include:

• Magnitude: Denoted as \(|\mathbf{u}|\), it is a measure of the "length" of the vector \(\mathbf{u}\).
• Direction: The orientation of the vector in space.
• Addition: Vectors can be added together by placing them head to tail. This is known as the triangle or parallelogram rule.
• Scalar multiplication: A vector can be multiplied by a scalar (a real number), which scales its magnitude without changing its direction.
• Zero Vector: A vector with zero magnitude and no specific direction.

Understanding these basic vector properties is crucial for solving problems in vector calculus and for working effectively with more complex operations like the cross and dot product.

Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space, denoted by \(\mathbf{u} \times \mathbf{v}\).

Important aspects of the cross product include:

• Result: The cross product results in a vector that is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\).
• Magnitude: The magnitude of \(\mathbf{u} \times \mathbf{v}\) is given by \(|\mathbf{u}||\mathbf{v}|\sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\).
• Direction: Determined by the right-hand rule, which states that if the fingers of your right hand curl from \(\mathbf{u}\) to \(\mathbf{v}\), your thumb points in the direction of the cross product.
• Anti-commutativity: \(\mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u})\).
• Distributive over addition: \(\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}\).
• Zero Vector: If \(\mathbf{u}\) and \(\mathbf{v}\) are parallel (or one is zero), the result is the zero vector \(\mathbf{0}\).

These properties make the cross product especially useful in physics for finding torques and rotational forces.

Dot Product

The dot product, or scalar product, is an algebraic operation that takes two vectors and returns a scalar.

Key features of the dot product are:

• Calculation: For vectors \(\mathbf{u}\) and \(\mathbf{v}\), the dot product is computed as \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos(\theta)\), where \(\theta\) is the angle between the vectors.
• Result: It returns a scalar value representing the "projection" of one vector on another.
• Perpendicularity: Two vectors are perpendicular if their dot product is zero because \(\cos(90^\circ) = 0\).
• Self Product: The dot product of a vector with itself gives the square of its magnitude: \(\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2\).
• Commutativity: \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\).
• Distributive Property: Over addition, meaning \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\).

Understanding the dot product is critical for applications in physics, projecting forces, and calculating work done by a force.

Vector Identities

Vector identities are equations involving vectors, holding under various operations like addition, dot product, and cross product.

Some essential vector identities are:

• Triangle Law of Addition: For any vectors \(\mathbf{u}\) and \(\mathbf{v}\), \(\mathbf{u} + \mathbf{v}\) forms the third side of a triangle with vertices at the tail of \(\mathbf{u}\) and \(\mathbf{v}\).
• Parallelogram Law: The vector sum \(\mathbf{u} + \mathbf{v}\) diagonally bisects a parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\).
• Distributive Laws: \(\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}\) and \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\).
• Orthogonal Complements: If \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = 0\), then \(\mathbf{w}\) is orthogonal to \(\mathbf{u} \times \mathbf{v}\).
• Triple Product Expansion: Also known as Lagrange's identity, \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\).

These identities are vital for simplifying vector expressions and proving more complex mathematical statements in vector calculus.