91Ó°ÊÓ

To grasp the notion of vector orthogonality, we must first understand the dot product, a central concept in linear algebra. The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. Let's dive into the dot product with an example.
The dot product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \) is calculated by multiplying corresponding components and summing the results: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \). This operation is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
What makes the dot product special? It can reveal the relationship between vectors, particularly, whether they're orthogonal. If the dot product is zero, it tells us that the vectors are perpendicular to each other in euclidean space. It's a fundamental tool that turns geometrical intuition into a crisp, algebraic criterion.

Linear algebra is a significant field within mathematics that delves into vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also fundamental to topics in applied mathematics, physics, and engineering.
Understanding vectors and operations like the dot product becomes simpler with knowledge of linear algebra concepts. Regarding our textbook exercise, linear algebra teaches us how to determine vector characteristics, such as direction and magnitude, and to solve for unknowns within vector equations. In this case, linear algebraic equations are used to capture the conditions for orthogonality among vectors.

Orthogonal vectors are a pair of vectors that meet at a right angle. In a three-dimensional space, this means they have no projection on each other—think of it as their paths never crossing. Our exercise specifically asks us to find all vectors orthogonal to a given vector \( \mathbf{u} \).
Using the definition from linear algebra, we employ the dot product to find all vectors \( \mathbf{v} \) that satisfy the criterion for orthogonality. We set up the equation \( \mathbf{u} \cdot \mathbf{v} = 0 \) and solve for \( \mathbf{v} \) to find all possible solutions. The beautiful thing about orthogonality is that it allows for infinite solutions; in this case, any vector \( (x, 4x, z) \) with any real numbers for \( x \) and \( z \) would be orthogonal to \( \mathbf{u} = (4, -1, 0) \). Here, \( y = 4x \) and \( z \) can be any value, symbolizing a whole line of solutions (a subspace) that are orthogonal to \( \mathbf{u} \).