Math Problem Statement

If (a, b, c) is the inverse element of (1, 2, 5) ∈ ℝ^3 of the vector space ℝ^3, then find the value of a - b + c.

Solution

The problem states that (a,b,c)(a, b, c) is the inverse element of the vector (1,2,5)(1, 2, 5) in R3\mathbb{R}^3, the real 3-dimensional vector space. To find the value of ab+ca - b + c, we need to determine the inverse of the given vector.

In a vector space, the inverse of a vector (x,y,z)(x, y, z) is the vector (x,y,z)(-x, -y, -z) because adding them results in the zero vector (0,0,0)(0, 0, 0).

Thus, the inverse of (1,2,5)(1, 2, 5) is (1,2,5)(-1, -2, -5). Therefore, we have: a=1,b=2,c=5.a = -1, \quad b = -2, \quad c = -5.

Now, we compute ab+ca - b + c: ab+c=(1)(2)+(5)=1+25=4.a - b + c = (-1) - (-2) + (-5) = -1 + 2 - 5 = -4.

Thus, the value of ab+ca - b + c is 4-4.

Would you like more details on the concept of inverse elements in vector spaces? Here are some related questions to expand your understanding:

  1. What is the general formula for finding the inverse of a vector in any vector space?
  2. How do inverse elements behave under vector addition?
  3. Can you explain the properties of vector spaces, particularly the identity and inverse elements?
  4. How is the dot product of vectors related to the inverse of a vector?
  5. What are some real-world applications of inverse elements in vector spaces?

Tip: In any vector space, the inverse of a vector is simply the negation of each of its components.

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Math Problem Analysis

Mathematical Concepts

Vector Space
Inverse Elements
Vector Addition

Formulas

Inverse of a vector (x, y, z) is (-x, -y, -z)

Theorems

Properties of Vector Spaces
Inverse Elements in Vector Spaces

Suitable Grade Level

Grades 10-12