Question

Дана правильная четырёхугольная пирамида KABCD, все рёбра которой равны 8 ед. изм. На рёбрах KC и KD соответственно отмечены серединные точки M и N. Определи косинус угла α между прямыми AN и DM.

Answer 1

Ответ: cosN = $\frac{1}{6}$

Объяснение:

Answer 2

Given Information:

We are given a regular quadrilateral pyramid KABCD, where all the edges are equal to 8 units. The midpoint of the edges KC and KD are marked as M and N, respectively. We need to determine the cosine of the angle α between the lines AN and DM.

Solution:

To find the cosine of the angle α, we need to find the dot product of the vectors AN and DM, and then divide it by the product of their magnitudes.

Let's start by finding the vectors AN and DM.

The vector AN can be obtained by subtracting the coordinates of point A from point N. Similarly, the vector DM can be obtained by subtracting the coordinates of point D from point M.

Next, we calculate the dot product of the vectors AN and DM.

Finally, we divide the dot product by the product of the magnitudes of the vectors AN and DM to find the cosine of the angle α.

Let's calculate the cosine of the angle α step by step.

Calculation:

1. Finding the vectors AN and DM:

The coordinates of point A are (0, 0, 0) and the coordinates of point N are (8, 0, 0). Therefore, the vector AN is (8, 0, 0).

The coordinates of point D are (0, 0, 8) and the coordinates of point M are (4, 0, 8). Therefore, the vector DM is (4, 0, 0).

2. Calculating the dot product of the vectors AN and DM:

The dot product of two vectors can be calculated by multiplying their corresponding components and summing them up.

The dot product of AN and DM is calculated as follows:

AN · DM = (8 * 4) + (0 * 0) + (0 * 0) = 32

3. Calculating the magnitudes of the vectors AN and DM:

The magnitude of a vector can be calculated using the formula: √(x^2 + y^2 + z^2), where (x, y, z) are the components of the vector.

The magnitude of vector AN is calculated as follows:

|AN| = √(8^2 + 0^2 + 0^2) = √64 = 8

The magnitude of vector DM is calculated as follows:

|DM| = √(4^2 + 0^2 + 0^2) = √16 = 4

4. Calculating the cosine of the angle α:

The cosine of an angle can be calculated using the formula: cos(α) = (AN · DM) / (|AN| * |DM|)

Substituting the values we calculated earlier:

cos(α) = 32 / (8 * 4) = 32 / 32 = 1

Answer:

The cosine of the angle α between the lines AN and DM is 1.

Please let me know if you need any further clarification or assistance.