What is the formula of \[{x^2} + {y^2} + {z^2}?\

The expression \[{x^2} + {y^2} + {z^2}\] represents the sum of the squares of three variables: \(x\), \(y\), and \(z\). This formula is commonly encountered in various fields of mathematics, particularly in geometry and algebra. It can be interpreted in different contexts, such as calculating distances in three-dimensional space or analyzing quadratic forms.

Understanding the Components

Each term in the expression \[{x^2}\], \[{y^2}\], and \[{z^2}\] is the square of a variable. Squaring a number means multiplying it by itself. For example:

• If \(x = 3\), then \({x^2} = 3 \times 3 = 9\).
• If \(y = 4\), then \({y^2} = 4 \times 4 = 16\).
• If \(z = 5\), then \({z^2} = 5 \times 5 = 25\).

When you add these squared values together, you get:

\[{x^2} + {y^2} + {z^2} = 9 + 16 + 25 = 50\]

Geometric Interpretation

In a three-dimensional Cartesian coordinate system, the expression \[{x^2} + {y^2} + {z^2}\] can be used to calculate the square of the distance from the origin (0, 0, 0) to a point \((x, y, z)\). The distance \(d\) from the origin to the point is given by the formula:

\[d = \sqrt{x^2 + y^2 + z^2}\]

Thus, \[{x^2} + {y^2} + {z^2}\] represents the squared distance, which is useful in various applications, including physics and engineering, where distances need to be calculated without taking the square root for simplification.

Applications in Mathematics

This expression also appears in the context of quadratic equations and optimization problems. For instance, if you want to minimize or maximize a function involving these variables, understanding how they interact through their squares is crucial. The expression can also be part of more complex formulas, such as in the Pythagorean theorem extended to three dimensions.

Example in Optimization

Consider a scenario where you want to find the minimum value of the function:

\[f(x, y, z) = x^2 + y^2 + z^2\]

To minimize this function, you would typically set the partial derivatives with respect to \(x\), \(y\), and \(z\) to zero and solve the resulting equations. This process helps in finding the point closest to the origin in three-dimensional space.

Summary of Key Points

• The expression \[{x^2} + {y^2} + {z^2}\] sums the squares of three variables.
• It can represent the squared distance from the origin to a point in 3D space.
• This formula is applicable in various mathematical contexts, including geometry and optimization.

Understanding this expression opens the door to deeper insights in both theoretical and applied mathematics. Whether you're working on geometry problems or exploring optimization techniques, the sum of squares is a fundamental concept that plays a significant role in many areas of study.