Theorem 104 Every vector in space can be written in the form ai+bj+ck in one and only one way.The numbers a,b , and c are called the first, second and third components respectively, of the vector.

This theorem is a fundamental concept in linear algebra and vector spaces. It essentially states that any vector in a three-dimensional space can be uniquely represented as a linear combination of three linearly independent vectors, which are typically chosen as the standard basis vectors:* **i:** The unit vector along the x-axis (1, 0, 0)* **j:** The unit vector along the y-axis (0, 1, 0)* **k:** The unit vector along the z-axis (0, 0, 1)**Explanation:**1. **Linear Combination:** The expression "ai + bj + ck" represents a linear combination of the basis vectors i, j, and k. The coefficients a, b, and c are scalar values that determine the magnitude and direction of the vector.2. **Uniqueness:** The theorem states that this representation is unique. This means that for any given vector, there is only one possible set of coefficients (a, b, c) that will accurately represent it.**Example:**Let's say you have a vector with coordinates (2, 3, -1). This vector can be written as:2i + 3j - kHere, a = 2, b = 3, and c = -1. These coefficients uniquely define the vector.**Why is this important?*** **Representation:** This theorem allows us to represent any vector in a simple and consistent way using a set of three numbers.* **Operations:** It simplifies vector operations like addition, subtraction, and scalar multiplication.* **Basis:** The concept of a basis is crucial in understanding vector spaces and linear transformations.**In summary:** Theorem 104 establishes the fundamental principle that any vector in three-dimensional space can be uniquely expressed as a linear combination of the standard basis vectors. This representation is essential for understanding and manipulating vectors in various mathematical and physical applications.