Question

Дано прямокутний паралелепіпед ABCDA1B1C1D1. Доведіть, що пряма ДД1 перпендикулярна до площини грані АВСД

Answer 1

Ответ:

Доказано!

Объяснение:

• Если прямая перпендикулярна двум пересекающимся прямым в плоскости, то она перпендикулярна этой плоскости.

Так как AD и DC∈(ABCD) , AD∩DC=D, ещё в прямоугольном параллелипипеде DD₁⊥AD и ⊥DC , то DD₁⊥(ABCD). Что и требовалось доказать!

Answer 2

Proof that line DD1 is perpendicular to the plane of face ABCD

To prove that line DD1 is perpendicular to the plane of face ABCD, we can use the concept of vectors and dot product.

Let's assume that the coordinates of point D are (x1, y1, z1), the coordinates of point D1 are (x2, y2, z2), and the coordinates of points A, B, C, and D are known.

To show that line DD1 is perpendicular to the plane of face ABCD, we need to show that the vector DD1 is orthogonal (perpendicular) to any vector lying in the plane of face ABCD.

The vector DD1 can be represented as:

DD1 = (x2 - x1, y2 - y1, z2 - z1)

Now, let's consider any vector lying in the plane of face ABCD. We can choose two vectors in the plane, such as AB and AD.

The vector AB can be represented as:

AB = (x_B - x_A, y_B - y_A, z_B - z_A)

Similarly, the vector AD can be represented as:

AD = (x_D - x_A, y_D - y_A, z_D - z_A)

To prove that line DD1 is perpendicular to the plane of face ABCD, we need to show that the dot product of vector DD1 with any vector lying in the plane of face ABCD is zero.

The dot product of two vectors is given by the formula:

Dot Product = (x1 * x2) + (y1 * y2) + (z1 * z2)

Now, let's calculate the dot product of vector DD1 with vector AB:

Dot Product_AB = (x1 * (x_B - x_A)) + (y1 * (y_B - y_A)) + (z1 * (z_B - z_A))

Similarly, let's calculate the dot product of vector DD1 with vector AD:

Dot Product_AD = (x1 * (x_D - x_A)) + (y1 * (y_D - y_A)) + (z1 * (z_D - z_A))

If both Dot Product_AB and Dot Product_AD are equal to zero, it means that line DD1 is perpendicular to the plane of face ABCD.

Please note that the specific coordinates of points A, B, C, D, D1, and the values of x1, y1, z1, x2, y2, z2 need to be known to perform the calculations and prove the perpendicularity.

Unfortunately, the search results did not provide specific information or formulas to directly prove the perpendicularity of line DD1 to the plane of face ABCD. Therefore, I am unable to provide a more detailed proof at this time.

If you have any further questions or need clarification, please let me know!