Problem 27 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Analyze a

We are given \(|\mathbf{u}| = \sqrt{\mathbf{u} \cdot \mathbf{u}}\). This is always true because the magnitude of a vector \(\mathbf{u}\) is defined as \(|\mathbf{u}| = \sqrt{\mathbf{u} \cdot \mathbf{u}}\).

Step 2: Analyze b

We need to determine if \(\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|\). This is not always true. While \(\mathbf{u} \cdot \mathbf{u}\) equals \(|\mathbf{u}|^2\), it is not equal to \(|\mathbf{u}|\) unless \(|\mathbf{u}|\) is 1.

Step 3: Analyze c

The expression \(\mathbf{u} \times \mathbf{0} = \mathbf{0} \times \mathbf{u} = \mathbf{0}\) is always true. The cross product of any vector with the zero vector is always the zero vector.

Step 4: Analyze d

For \(\mathbf{u} \times (-\mathbf{u}) = \mathbf{0}\), this is always true. The cross product of a vector with its negative results in the zero vector.

Step 5: Analyze e

The equality \(\mathbf{u} \times \mathbf{v} = \mathbf{v} \times \mathbf{u}\) is not always true. The cross product is anti-commutative: \(\mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u})\).

Step 6: Analyze f

The equation \(\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}\) is always true. This follows the distributive property of the cross product over addition.

Step 7: Analyze g

The relation \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{v} = 0\) is always true because \(\mathbf{u} \times \mathbf{v}\) is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\).

Step 8: Analyze h

The equation \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) is always true. This is a property known as the scalar triple product identity.

Key Concepts

Vector Operations

Vectors are fundamental in physics and mathematics for representing quantities that have both magnitude and direction. The basic operations involving vectors include addition, subtraction, and scalar multiplication. These operations lay the groundwork for understanding more complex concepts in vector calculus.

• Addition: To add two vectors, place them tail-to-tail and draw the resultant vector from the free tail to the free head.
• Subtraction: To subtract, add the negative of the vector you want to subtract.
• Scalar Multiplication: Multiplying a vector by a scalar changes the magnitude of the vector without altering its direction, unless the scalar is negative, which also reverses its direction.

Understanding these basic operations allows us to progress to more advanced operations like the cross and dot products, which are essential for vector calculus.

Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. This operation results in a vector that is perpendicular to both of the original vectors.

The magnitude of the cross product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:\[|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin\theta\]where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \). The direction of the cross product is determined using the right-hand rule.

Some key properties of the cross product include:

• Anti-Commutativity: \( \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}) \)
• Distributive Property: \( \mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} \)
• Zero Result: The cross product of a vector with itself or the zero vector is always the zero vector.

Being very different from the dot product, the cross product is essential in physics for calculations involving torque and rotational motion.

Scalar Triple Product

The scalar triple product involves three vectors and results in a scalar. It is defined using both the dot and the cross products. If you have vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \), the scalar triple product is given by:\[(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}\]or equivalently by the identity:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\]This product computes the volume of the parallelepiped formed by the three vectors. A useful property of the scalar triple product is that it does not change when the order of the cross and dot products are swapped, adhering to the identity above.

However, it changes sign when the vectors are cyclically permuted, indicating the directed volume and orientation of the parallelepiped are significant.

• If the scalar triple product is zero, it implies that the vectors are coplanar.
• It helps in computations involving flux in multivariable calculus and physics.

Dot Product

The dot product, also known as the scalar product, results in a scalar and measures the magnitude of the projection of one vector onto another. For vectors \( \mathbf{a} \) and \( \mathbf{b} \), the dot product is defined as:\[\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos\theta\]where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \).

It has several important properties:

• Commutative: \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \)
• Distributive over addition: \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \)
• Orthogonality: If \( \mathbf{a} \cdot \mathbf{b} = 0 \), the vectors are orthogonal.

This operation is key in projections, determining angles between vectors, and simplifying the work involved in physics and engineering computations. It's a straightforward yet powerful tool in vector calculus.