Question

Найти расстояние от точки М (2;3;1) до координатных площин.

Answer 1

Расстояние от точки М до плоскости хоу равно z -координате точки М, т.е. 1.

Расстояние от точки М до плоскости уоz равно х -координате точки М, т.е. 2.

Расстояние от точки М до плоскости хоz равно у -координате точки М, т.е. 3.

 

Answer 2

Finding the Distance from Point M to Coordinate Planes

To find the distance from point M(2, 3, 1) to the coordinate planes, we need to calculate the perpendicular distance from the point to each plane. The coordinate planes are the xy-plane, xz-plane, and yz-plane.

Let's calculate the distance to each plane:

1. Distance to the xy-plane: The xy-plane is the plane where z = 0. To find the distance from point M to the xy-plane, we need to find the perpendicular distance from point M to the plane z = 0.

The formula to calculate the distance between a point and a plane is: ``` distance = |Ax + By + Cz + D| / sqrt(A^2 + B^2 + C^2) ``` In this case, the equation of the plane is z = 0, which can be written as 0x + 0y + 1z + 0 = 0. Therefore, A = 0, B = 0, C = 1, and D = 0.

Plugging in the values, we get: ``` distance = |0*2 + 0*3 + 1*1 + 0| / sqrt(0^2 + 0^2 + 1^2) = |1| / sqrt(1) = 1 / 1 = 1 ``` So, the distance from point M to the xy-plane is 1 unit.

2. Distance to the xz-plane: The xz-plane is the plane where y = 0. To find the distance from point M to the xz-plane, we need to find the perpendicular distance from point M to the plane y = 0.

Using the same formula as above, the equation of the plane is y = 0, which can be written as 0x + 1y + 0z + 0 = 0. Therefore, A = 0, B = 1, C = 0, and D = 0.

Plugging in the values, we get: ``` distance = |0*2 + 1*3 + 0*1 + 0| / sqrt(0^2 + 1^2 + 0^2) = |3| / sqrt(1) = 3 / 1 = 3 ``` So, the distance from point M to the xz-plane is 3 units.

3. Distance to the yz-plane: The yz-plane is the plane where x = 0. To find the distance from point M to the yz-plane, we need to find the perpendicular distance from point M to the plane x = 0.

Using the same formula as above, the equation of the plane is x = 0, which can be written as 1x + 0y + 0z + 0 = 0. Therefore, A = 1, B = 0, C = 0, and D = 0.

Plugging in the values, we get: ``` distance = |1*2 + 0*3 + 0*1 + 0| / sqrt(1^2 + 0^2 + 0^2) = |2| / sqrt(1) = 2 / 1 = 2 ``` So, the distance from point M to the yz-plane is 2 units.

Therefore, the distances from point M(2, 3, 1) to the coordinate planes are as follows: - Distance to the xy-plane: 1 unit - Distance to the xz-plane: 3 units - Distance to the yz-plane: 2 units.

Please note that the above calculations are based on the assumption that the coordinate planes are perpendicular to each other and that the point M is not on any of the coordinate planes.

I hope this helps! Let me know if you have any further questions.