Problem 28 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Analyzing Statement a

Statement a is \( \mathbf{a} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \). The dot product \( \mathbf{a} \cdot \mathbf{v} \) equals \( \mathbf{v} \cdot \mathbf{u} \) only if \( \mathbf{a} = \mathbf{u} \). Since \( \mathbf{a} \) and \( \mathbf{u} \) can be different vectors, this statement is not always true.

Step 2: Analyzing Statement b

Statement b is \( \mathbf{u} \times \mathbf{v} = - ( \mathbf{v} \times \mathbf{u} ) \). This is always true because the cross product is anticommutative, meaning if you swap the operands, the result is the negative of the original cross product.

Step 3: Analyzing Statement c

Statement c is \( (-\mathbf{u}) \times \mathbf{v} = -(\mathbf{u} \times \mathbf{v}) \). This is always true because multiplying one vector by \(-1\) in the cross product results in negating the entire cross product.

Step 4: Analyzing Statement d

Statement d is \((c \mathbf{u}) \cdot \mathbf{v} = \mathbf{u} \cdot (c \mathbf{v}) = c(\mathbf{u} \cdot \mathbf{v}) \). This is always true because the dot product is distributive over scalar multiplication.

Step 5: Analyzing Statement e

Statement e is \( c(\mathbf{u} \times \mathbf{v}) = (c \mathbf{u}) \times \mathbf{v} = \mathbf{u} \times (c \mathbf{v}) \). This is not always true because scalar multiplication of the cross product with respect to one vector does not follow the same properties as the dot product in all situations.

Step 6: Analyzing Statement f

Statement f is \( \mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^{2} \). This is always true because the dot product of a vector with itself gives the magnitude squared of that vector.

Step 7: Analyzing Statement g

Statement g is \((\mathbf{u} \times \mathbf{u}) \cdot \mathbf{u} = 0 \). This is always true because the cross product of any vector with itself is the zero vector, and the dot product of the zero vector with any vector is zero.

Step 8: Analyzing Statement h

Statement h is \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u} = \mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) \). This is not always true because the left side represents the scalar triple product which evaluates to zero, whereas the right side does not generally equal the same unless certain conditions are met.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It involves two vectors and results in a scalar. If you have two vectors, say \( \mathbf{a} \) and \( \mathbf{b} \), the dot product is calculated as:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \] where \( a_1, a_2, \) and \( a_3 \) are components of \( \mathbf{a} \) and \( b_1, b_2, \) and \( b_3 \) are components of \( \mathbf{b} \).The result is a single number (a scalar), not a vector.
This operation measures how much one vector "goes in the direction" of another.
When the dot product is zero, it indicates that the vectors are orthogonal, meaning they are at a 90-degree angle to each other.Additionally, the dot product has a commutative property, meaning that \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
This property can sometimes cause confusion in problems if both vectors are not explicitly the same, highlighting the importance of context in vector calculations.

Cross Product

The cross product, or vector product, is another key operation in vector algebra. Unlike the dot product, the cross product of two vectors results in another vector that is perpendicular to the plane containing the original vectors.
Given vectors \( \mathbf{u} \) and \( \mathbf{v} \), the cross product is expressed as:\[ \mathbf{u} \times \mathbf{v} = |\mathbf{u}||\mathbf{v}|\sin\theta\ \mathbf{n} \] where \( |\mathbf{u}| \) and \( |\mathbf{v}| \) are the magnitudes of the vectors, \( \theta \) is the angle between them, and \( \mathbf{n} \) is the unit vector perpendicular to both.The result is a vector not a scalar, emphasizing its difference from the dot product.
The magnitude of the cross product gives the area of the parallelogram formed by the two vectors.
The cross product is not commutative but follows an important property known as being anticommutative.

Scalar Multiplication

Scalar multiplication in vector algebra involves multiplying each component of a vector by a number (a scalar). This operation changes the magnitude of the vector but not its direction, unless the scalar is negative, in which case the direction is reversed.For a vector \( \mathbf{v} = (v_1, v_2, v_3) \) and a scalar \( c \), the scalar multiplication is defined as:\[ c \mathbf{v} = (cv_1, cv_2, cv_3) \]The operation is straightforward and essential in many applications involving vectors.
It is distributive over vector addition and compatible with multiplication, meaning \( c(d \mathbf{v}) = (cd) \mathbf{v} \).
Such properties are crucial in vector spaces and linear algebra operations.

Anticommutative Property

The anticommutative property is a significant aspect relating to the cross product in vector algebra. It indicates that changing the order of the vectors in the cross product negates the result.In mathematical terms, for vectors \( \mathbf{u} \) and \( \mathbf{v} \), the anticommutative property is defined as:\[ \mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u}) \]This property is particularly important as it differentiates the behavior of the cross product from the dot product, which is commutative.Hence, when solving vector problems involving cross products, this property helps to ensure the correct direction of the resulting vector.
In practical applications, understanding this property aids in analyzing forces in physics and engineering contexts where orientation matters.