Final answer:
The distance from the point (2, -6, 5) to the origin, another point, a plane, and a line are calculated using various formulas specific to the object in 3D space.
Step-by-step explanation:
To find the distance from a point to various objects in three-dimensional space, different formulas are used depending on the object. The distance from the point (2, −6, 5) to other objects can be found as follows:
- Distance to the origin is found by applying the distance formula for points in 3D space. Since the origin is at (0, 0, 0), the distance is √((2-0)² + (−6-0)² + (5-0)²).
- Distance to another point, (1, -2, 4), is also found using the distance formula: √((2-1)² + (−6+2)² + (5-4)²).
- Distance to a plane given by the equation x + 2y - z = 0 is found using the formula: |Ax1 + By1 + Cz1 + D| / √(A² + B² + C²), substituting the point's coordinates into this formula and the coefficients from the plane's equation.
- Distance to a line passing through (3, 1, -1) and (4, -3, 2) requires finding the equation of the line, then using a formula for the shortest distance from a point to a line in 3D, which is based on vector projections and cross products.