Question

Найдите угол между диагоналями параллелограмма, если заданы три его вершины A(2;1; 3) , B(5; 2;−1) , C(−3; 3;−3) .Через векторы решается как то

Answer 1

Даны точки А(2;1;3) В(5;2;-1) С(-3;3;-3)

Сторонами параллелограмма будут АВ, ВС, СD, DA

Найдем диагональ АС:

$AC=(-3-2;3-1;-3-3)=(-5;2;6)$

Координаты точки D(x;y;z)

Надо найти диагональ BD

Диагонали пересекаются и точкой пересечения делятся пополам:
пусть точкой пересечения будет точка О, тогда координаты точки О (на диагонали АС:

$O( \frac{2-3}{2}; \frac{1+3}{2}; \frac{3-3}{2})=(- \frac{1}{2};2;0)$

с другой стороны координатами точки O  (на диагонали BD) будут:

$O( \frac{5+x}{2}; \frac{2+y}{2}; \frac{-1+z}{2})$

приравняем и найдем координаты точки D

$\frac{5+x}{2}=- \frac{1}{2} 5+x=-1 x=-6$

$\frac{2+y}{2}=2 y=2$

$\frac{-1+z}{2}=0 z=1$

таким образом D(-6;3;1)

теперь найдем BD

$BD(-6-5;2-2;1+1)=(-11;0;2)$

найдем длину |AC| и |BD|

$|AC|= \sqrt{25+4+36}= \sqrt{65}$

$|BD|= \sqrt{121+4}= \sqrt{125}$

найдем угол

$Cos \alpha = \frac{AC*BD}{|AC|*|BD|}$

$Cos \alpha = \frac{(-11*(-5)+0*2+2*(-6))}{ \sqrt{65}* \sqrt{125}}= \frac{55-12}{25 \sqrt{13} }= \frac{43}{25 \sqrt{13}}$

Answer 2

Finding the Angle Between the Diagonals of a Parallelogram

To find the angle between the diagonals of a parallelogram using vectors, we can follow these steps:

1. Find the Vectors for the Diagonals: - Given three vertices A(2, 1, 3), B(5, 2, -1), and C(-3, 3, -3), we can find the vectors for the diagonals using the formula: - For the diagonal AC, the vector is AC = C - A - For the diagonal BD, the vector is BD = D - B

2. Calculate the Angle: - Once we have the vectors for the diagonals, we can use the formula to find the angle between them: - cos(θ) = (u . v) / (|u| * |v|) - Where u and v are the vectors for the diagonals, and θ is the angle between them.

Let's proceed with these steps to find the angle between the diagonals of the parallelogram.

Calculating the Vectors for the Diagonals

The vectors for the diagonals can be calculated as follows: - For the diagonal AC: AC = C - A - For the diagonal BD: BD = D - B

Calculating the Angle Between the Diagonals

Using the formula cos(θ) = (u . v) / (|u| * |v|), where u and v are the vectors for the diagonals, we can calculate the angle between the diagonals.

This approach will help us find the angle between the diagonals of the parallelogram using the given vertices.