Question

Помогите найти расстояние от точки М (3;5;5) до прямой х-1/(-2)=у/4=z/1

Answer 1

Решение во вложенном файле

Answer 2

Finding the Distance from a Point to a Line in 3D Space

To find the distance from a point to a line in 3D space, we can use vector projection. In this case, we want to find the distance from point M(3, 5, 5) to the line defined by the equation x-1/(-2) = y/4 = z/1.

Step 1: Find a vector that points along the line

To find a vector that points along the line, we can consider the direction ratios of the line. From the given equation, we can see that the direction ratios of the line are (-2, 4, 1). We can use this vector as a direction vector that points along the line.

Step 2: Find a vector connecting the point to the line

To find a vector connecting the point M(3, 5, 5) to the line, we need to find a vector that starts at M and is perpendicular to the line. Let's call this vector N.

We can find vector N by subtracting the coordinates of a point on the line from the coordinates of point M. Let's take the point P(1, 0, 0) on the line.

N = M - P = (3, 5, 5) - (1, 0, 0) = (2, 5, 5)

Step 3: Calculate the distance

The distance from point M to the line can be calculated using the formula:

distance = |N| / |direction vector|

where |N| represents the magnitude of vector N and |direction vector| represents the magnitude of the direction vector of the line.

To find the magnitude of a vector, we can use the formula:

|V| = sqrt(Vx^2 + Vy^2 + Vz^2)

where Vx, Vy, and Vz are the components of the vector.

Let's calculate the magnitude of vector N and the magnitude of the direction vector:

|N| = sqrt(2^2 + 5^2 + 5^2) = sqrt(54) = 3sqrt(6)

|direction vector| = sqrt((-2)^2 + 4^2 + 1^2) = sqrt(21)

Now, we can calculate the distance:

distance = |N| / |direction vector| = (3sqrt(6)) / sqrt(21) = 3sqrt(6) / sqrt(21) * sqrt(21) / sqrt(21) = 3sqrt(126) / 21 = sqrt(126) / 7

Therefore, the distance from the point M(3, 5, 5) to the line x-1/(-2) = y/4 = z/1 is sqrt(126) / 7.