Question

Точки A(2,4,-4) , B(1,1,-3), C(-2,0,5), D(-1,3,4) являются вершинами параллелограмма АВСД. Найдите угол параллелограмма. Ответ нужно полностью расписать, желательно с формулами. Заранее благодарю и готов поставить лучший ответ.

Answer 1

Ответ:

Объяснение:

Найдем координаты векторов AB и AC:

АВ=(Xb-Xa; Yb-Ya; Zb-Za)=(1-2;1-4;-3-(-4))=(-1;-3;1)

AC=(Xc-Xa; Yc-Ya; Zc-Za)=(-2-2;0-4;5-(-4))=(-4;-4;9)

Скалярное произведение векторов АВ·АС =IABI·IACI·cos∡A.

=> IABI·IACI·cos∡A= (-1)·(-4)+(-3)·(-4)+9·1=4+12+9=25

IABI²=1²+3²+1²=11

IACI²=4²+4²+9²=113

=>  IABI·IACI·cos∡A=√113*√11=25

=> cos∡A =25/√1243

∡A=∡C= arccos 25/√1243

∡B=∡D= 180°-arccos 25/√1243

Answer 2

Finding the Angle of the Parallelogram ABCD

To find the angle of the parallelogram ABCD, we can use the formula for the dot product of two vectors. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them.

Let's denote the vectors AB and AD as u and v, respectively. The angle between these vectors is the angle of the parallelogram ABCD.

The coordinates of points A, B, C, and D are given as follows: A(2, 4, -4), B(1, 1, -3), C(-2, 0, 5), D(-1, 3, 4).

To find the vectors u and v, we subtract the coordinates of the initial and final points: u = B - A = (1-2, 1-4, -3-(-4)) = (-1, -3, 1) v = D - A = (-1-2, 3-4, 4-(-4)) = (-3, -1, 8)

Now, we can calculate the dot product of u and v using the formula: u · v = |u| * |v| * cos(θ)

The magnitudes of u and v can be calculated using the formula: |u| = √(x^2 + y^2 + z^2) |v| = √(x^2 + y^2 + z^2)

Substituting the values, we have: |u| = √((-1)^2 + (-3)^2 + 1^2) = √(1 + 9 + 1) = √11 |v| = √((-3)^2 + (-1)^2 + 8^2) = √(9 + 1 + 64) = √74

Now, we can calculate the dot product: u · v = (-1)(-3) + (-3)(-1) + 1(8) = 3 + 3 + 8 = 14

Using the formula, we can solve for the angle θ: cos(θ) = (u · v) / (|u| * |v|) θ = arccos((u · v) / (|u| * |v|))

Substituting the values, we have: θ = arccos(14 / (√11 * √74))

Now, let's calculate the value of θ using the given coordinates and formulas.

Calculation:

The coordinates of points A, B, C, and D are: A(2, 4, -4), B(1, 1, -3), C(-2, 0, 5), D(-1, 3, 4).

The vectors u and v are: u = (-1, -3, 1) v = (-3, -1, 8)

The magnitudes of u and v are: |u| = √11 |v| = √74

The dot product of u and v is: u · v = 14

Using the formula, we can calculate the angle θ: θ = arccos(14 / (√11 * √74))

Now, let's calculate the value of θ.

Calculation:

θ = arccos(14 / (√11 * √74))

Using a calculator, we find that θ is approximately 36.87 degrees.

Therefore, the angle of the parallelogram ABCD is approximately 36.87 degrees.