Understanding the Distance from a Point to the X-axis in 3D Space

In three-dimensional space, the distance of a point from the x-axis is determined by its y and z coordinates. Let's explore the detailed calculation for the point P(3,4,5).

Calculation of Distance of P(3,4,5) from the X-axis

The distance of a point from the x-axis is determined by its y-coordinate. For the point P(3,4,5), the y-coordinate is 4. Therefore, the distance of point P from the x-axis is 4 units.

Using the Pythagorean Theorem for 3D Space

In a three-dimensional space, the point P(3,4,5) can be visualized as moving upward 4 units and inwards 5 units from the x-axis. The x-axis, in 3D, has the equation 'y0, z0', meaning any point on it has a y and z value of 0.

To find the distance from point P(3,4,5) to the x-axis, we calculate the hypotenuse of the triangle formed by these upward and inwards movements using the Pythagorean Theorem:

sqrt(4^2 5^2) sqrt(16 25) sqrt(41)

Thus, the distance from point P(3,4,5) to the x-axis is the square root of 41, approximately 6.403 units.

Minimizing Distance to a Point on the X-axis

To find the point on the x-axis closest to P(3,4,5), we consider the distance formula:

sqrt[(3 - x)^2 4^2 5^2]

This simplifies to:

sqrt[(3 - x)^2 16 25] sqrt[(3 - x)^2 41]

The expression (3 - x)^2 will be minimized when x 3. Thus, the point on the x-axis closest to P(3,4,5) is (3,0,0), and the distance is:

sqrt[3^2 4^2 5^2] sqrt(9 16 25) sqrt(50) 7.071 units.

Conclusion

The distance from the point P(3,4,5) to the x-axis is the square root of 41, approximately 6.403 units. This calculation is essential for understanding the geometric relationships in three-dimensional space, particularly in applications involving distance measurements and spatial geometry.

Key Takeaways:

The distance from a point to the x-axis in 3D space is determined by its y and z coordinates. The Pythagorean Theorem is used to calculate distances in three-dimensional space. The point on the x-axis closest to a given point in 3D space is found by minimizing the distance formula.