Problem 40 - Algebra and Trigonometry

Question

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors, and let \(a\) be a scalar. Prove the given property. $$ (a \mathbf{u}) \cdot \mathbf{v}=a(\mathbf{u} \cdot \mathbf{v})=\mathbf{u} \cdot(a \mathbf{v}) $$

Step-by-Step Solution

Answer

The property holds: \((a \mathbf{u}) \cdot \mathbf{v} = a(\mathbf{u} \cdot \mathbf{v}) = \mathbf{u} \cdot (a \mathbf{v})\).

Step 1: Understand the Dot Product

The dot product of two vectors \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\) is defined as \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\). This operation results in a scalar.

Step 2: Scalar Multiplication and Dot Product

When a vector \(\mathbf{u}\) is multiplied by a scalar \(a\), the resulting vector is \(a \mathbf{u} = (au_1, au_2, au_3)\). The dot product of this with \(\mathbf{v}\) becomes \((a \mathbf{u}) \cdot \mathbf{v} = au_1v_1 + au_2v_2 + au_3v_3 = a(u_1v_1 + u_2v_2 + u_3v_3) = a(\mathbf{u} \cdot \mathbf{v})\).

Step 3: Dot Product with Scalar Multiplied \(\mathbf{v}\)

Similarly, for the vector \(a \mathbf{v} = (av_1, av_2, av_3)\), the dot product with \(\mathbf{u}\) is \(\mathbf{u} \cdot (a \mathbf{v}) = u_1(av_1) + u_2(av_2) + u_3(av_3) = a(u_1v_1 + u_2v_2 + u_3v_3) = a(\mathbf{u} \cdot \mathbf{v})\).

Step 4: Conclusion

By verifying the results in Steps 2 and 3, we conclude that \((a \mathbf{u}) \cdot \mathbf{v} = a(\mathbf{u} \cdot \mathbf{v}) = \mathbf{u} \cdot (a \mathbf{v})\). The given property is proven to be true.

Key Concepts

Vector Operations

Vector operations form the core of vector algebra, and they include basic operations like addition, subtraction, and dot products between vectors. Together, these operations allow us to manipulate vectors to solve problems in physics, engineering, and computer graphics, among other fields. The dot product, in particular, is a fundamental operation, combining two vectors to yield a scalar.
The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\), which is written as \(\mathbf{u} \cdot \mathbf{v}\), is calculated by multiplying corresponding components of the vectors and then summing these products. This operation results in a single scalar value:

• For \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\), the dot product is \(u_1v_1 + u_2v_2 + u_3v_3\).

The dot product is particularly useful in determining whether two vectors are perpendicular (i.e., their dot product is zero) and in projecting one vector onto another.

Scalar Multiplication

Scalar multiplication is an operation that involves multiplying a vector by a scalar, a process which alters the magnitude of the vector without changing its direction (unless the scalar is negative, in which case the direction is also reversed). It’s an important tool in vector algebra that scales vectors, making them longer or shorter as needed by multiplying each component by the scalar.
Consider a vector \(\mathbf{u} = (u_1, u_2, u_3)\) and a scalar \(a\). The result of scalar multiplication, \(a\mathbf{u}\), is a new vector with components \((au_1, au_2, au_3)\). This operation is vital in numerous applications, such as uniformly stretching a vector field or applying scalars to solutions of linear equations in matrix algebra.

Vector Algebra

Vector algebra encompasses a whole system of operations involving vectors, including both vector operations like addition, subtraction, and the cross product, as well as operations like scalar multiplication. Within this system, properties such as commutativity, distributivity, and associativity govern how these operations can be performed.
One of the interesting properties when dealing with vector algebra is the interaction between dot and scalar multiplication. The given property \((a \mathbf{u}) \cdot \mathbf{v} = a(\mathbf{u} \cdot \mathbf{v}) = \mathbf{u} \cdot (a \mathbf{v})\) showcases that multiplying a vector by a scalar before or after taking the dot product will yield the same result. This demonstrates the associative nature of scalar multiplication with respect to the dot product.

• It highlights the distributive law: \((a \mathbf{u}) \cdot \mathbf{v}\) distributes across the components of \(\mathbf{u}\) and \(\mathbf{v}\).
• The same scalar can either be factored out before or after performing the dot product without affecting the outcome.

Vector algebra and these foundational properties enable complex vector manipulations, which are essential for modeling and solving real-world problems in various scientific and engineering disciplines.