Problem 28 - Thomas Calculus

Question

Which of the following are always true, and which are not always true? Give reasons for your answers. \(\begin{array}{ll}{\text { a. } \mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}} & {\text { b. } \mathbf{u} \times \mathbf{v}=-(\mathbf{v} \times \mathbf{u})} \\ {\mathbf{c} \cdot(-\mathbf{u}) \times \mathbf{v}=-(\mathbf{u} \times \mathbf{v})} & {}\end{array}\) d. \((c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})=c(\mathbf{u} \cdot \mathbf{v}) \quad\) (any number \(c )\) e. \(c(\mathbf{u} \times \mathbf{v})=(c \mathbf{u}) \times \mathbf{v}=\mathbf{u} \times(c \mathbf{v}) \quad\) (any number \(c )\) \(\mathbf{f} . \mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|^{2} \quad\) g. \((\mathbf{u} \times \mathbf{u}) \cdot \mathbf{u}=0\) \(\mathbf{h} .(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u}=\mathbf{v} \cdot(\mathbf{u} \times \mathbf{v})\)

Step-by-Step Solution

Answer

True: a, b, d, f, g. Not always true: c, e, h.

Step 1: Analyze Property (a)

The property \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \) is the commutative property of dot product. This property is always true for any vectors \( \mathbf{u} \) and \( \mathbf{v} \). Dot product is commutative, so the order does not matter.

Step 2: Analyze Property (b)

The property \( \mathbf{u} \times \mathbf{v} = - ( \mathbf{v} \times \mathbf{u} ) \) is the anti-commutative property of cross product. This property is always true. The cross product of two vectors changes sign if the order of the vectors is reversed.

Step 3: Analyze Property (c)

The property \( \mathbf{c} \cdot (-\mathbf{u}) \times \mathbf{v} = - ( \mathbf{u} \times \mathbf{v} ) \) needs verification: Expanding \( \mathbf{c} \cdot (-\mathbf{u}) \times \mathbf{v} \) gives \( - ( \mathbf{c} \cdot (\mathbf{u} \times \mathbf{v}) ) \), not necessarily equal to \( - ( \mathbf{u} \times \mathbf{v} ) \). Thus, it is not always true.

Step 4: Analyze Property (d)

The property \( (c \mathbf{u}) \cdot \mathbf{v} = \mathbf{u} \cdot (c \mathbf{v}) = c (\mathbf{u} \cdot \mathbf{v}) \) shows the distribution of scalar multiplication over dot product. This is true for all vectors and scalar \( c \) due to the associativity of multiplication and distribution of scalars.

Step 5: Analyze Property (e)

The property \( c(\mathbf{u} \times \mathbf{v}) = (c \mathbf{u}) \times \mathbf{v} = \mathbf{u} \times (c \mathbf{v}) \) shows the distribution of scalar multiplication over cross product. This set of equations is false because while the first equality \( c(\mathbf{u} \times \mathbf{v}) = (c \mathbf{u}) \times \mathbf{v} \) is true, the second equality \( \mathbf{u} \times (c \mathbf{v}) \) generally results in a different vector.

Step 6: Analyze Property (f)

The property \( \mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2 \) is always true. The dot product of any vector \( \mathbf{u} \) with itself is the squared magnitude of the vector.

Step 7: Analyze Property (g)

The property \( (\mathbf{u} \times \mathbf{u}) \cdot \mathbf{u} = 0 \) is always true. The cross product of any vector with itself is the zero vector, so dotting it with \( \mathbf{u} \) yields zero.

Step 8: Analyze Property (h)

The property \( (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u} = \mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) \) can be rewritten using the scalar triple product identity as zero on both sides because the cross product produces a vector perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \). However, \( \mathbf{u} \cdot \mathbf{u} eq 0 \), thus this statement is not always true.

Key Concepts

Commutative Property

The commutative property is a fundamental characteristic in mathematics, especially in operations such as addition and multiplication, where the order of the terms does not change the result. In vector calculus, however, this property applies differently depending on the operation used. For instance, the dot product is commutative.

• The dot product of two vectors, \( \mathbf{u} \cdot \mathbf{v} \), is always equal to \( \mathbf{v} \cdot \mathbf{u} \).
• This property simplifies algebraic manipulations as the vectors can be reordered without affecting the result.

Understanding the commutative nature of the dot product helps in streamlining calculations involving vectors in physics and engineering, where direction and magnitude play essential roles.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. The geometric interpretation of the dot product involves multiplying the magnitudes of the two vectors and the cosine of the angle between them.

• Mathematically, if \( \theta \) is the angle between two vectors \( \mathbf{u} \) and \( \mathbf{v} \), their dot product is defined as \( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos{\theta} \).
• The result is a scalar, hence the alternative name "scalar product."
• The dot product is commutative, so \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \).

This property is frequently used in projecting one vector onto another and in computing vector magnitudes.

Cross Product

The cross product, or vector product, differs from the dot product by yielding a vector, not a scalar. This vector is orthogonal (perpendicular) to the original two vectors, making the cross product a cornerstone concept in geometry and physics.

• The result of \( \mathbf{u} \times \mathbf{v} \) is a vector perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).
• Its magnitude is \( |\mathbf{u}| |\mathbf{v}| \sin{\theta} \), where \( \theta \) is the angle between \( \mathbf{u} \) and \( \mathbf{v} \).
• The cross product is anti-commutative, meaning \( \mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u}) \).

This anti-commutative property means that swapping the order of two vectors changes the direction of the resulting vector to its opposite.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, stretching or shrinking the vector while maintaining its direction. It is crucial in vector spaces and finds uses in both dot and cross product operations.

• If \( c \) is a scalar and \( \mathbf{u} \) a vector, then \( c\mathbf{u} \) is a vector in the same direction as \( \mathbf{u} \) but scaled by the factor \( c \).
• For the dot product, \( (c \mathbf{u}) \cdot \mathbf{v} = \mathbf{u} \cdot (c \mathbf{v}) = c(\mathbf{u} \cdot \mathbf{v}) \).
• Regarding the cross product, \( c(\mathbf{u} \times \mathbf{v}) = (c\mathbf{u}) \times \mathbf{v} = \mathbf{u} \times (c\mathbf{v}) \).

By scaling vectors, scalar multiplication allows for adjustments in magnitude, a common operation in physics for achieving desired force vectors or transformations.