Question

55 БАЛЛОВ!!!!! В трапеции ABCD ∠ = 60°, ∠ = 60°, ⃗⃗⃗⃗⃗⃗ = 2, ⃗⃗⃗⃗⃗⃗ = 3. Найти ⃗⃗⃗⃗⃗⃗ ∙ ⃗⃗⃗⃗⃗⃗ Cделать рисунок и указать векторы

Answer 1

Ответ:

Пошаговое объяснение:

В этой типовой задаче неверное условие, 60 градусам равен угол D, а не угол В.

Если угол В равен 60 градусам, то:

1. Или  трапеция не получается, так как противолежащие  углы трапеции не могут быть острыми оба, или тупыми оба. (посмотрите рисунок на втором листе)

2. Или  основаниями становятся АВ и СD. Но и в этом случае, АС не может быть перпендикулярен основанию СD.

Answer 2

Problem Analysis

We are given a trapezoid ABCD with angles ∠A = 60°, ∠B = 60°, and vectors ⃗AB = 2 and ⃗CD = 3. We need to find the vector ⃗AC ⋅ ⃗BD and create a diagram indicating the vectors.

Solution

To find the vector ⃗AC ⋅ ⃗BD, we first need to find the vectors ⃗AC and ⃗BD.

Let's start by finding ⃗AC. Since ∠A = 60°, we can use the law of cosines to find the length of ⃗AC. The law of cosines states that for a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab cos(C)

In our case, ⃗AC is the diagonal of the trapezoid, and we can consider the trapezoid as a triangle with sides ⃗AB, ⃗BC, and ⃗AC. Using the law of cosines, we can find the length of ⃗AC:

⃗AC^2 = ⃗AB^2 + ⃗BC^2 - 2⃗AB⃗BC cos(∠B)

Since ∠B = 60° and ⃗AB = 2, we can substitute these values into the equation:

⃗AC^2 = 2^2 + ⃗BC^2 - 2(2)(⃗BC) cos(60°)

Simplifying the equation:

⃗AC^2 = 4 + ⃗BC^2 - 2⃗BC

Now, let's find ⃗BD. Since ⃗BD is parallel to ⃗AC, they have the same length. Therefore, ⃗BD = ⃗AC.

Now that we have ⃗AC and ⃗BD, we can find ⃗AC ⋅ ⃗BD by taking the dot product of the two vectors:

⃗AC ⋅ ⃗BD = |⃗AC| |⃗BD| cos(θ)

Since ⃗AC = ⃗BD, we can rewrite the equation as:

⃗AC ⋅ ⃗BD = |⃗AC|^2 cos(θ)

Substituting the values we found earlier:

⃗AC ⋅ ⃗BD = ⃗AC^2 cos(θ)

Now, let's calculate the value of ⃗AC ⋅ ⃗BD using the given information.

Calculation

Using the equation ⃗AC^2 = 4 + ⃗BC^2 - 2⃗BC, we can substitute the value of ⃗BC = 3:

⃗AC^2 = 4 + 3^2 - 2(3)

Simplifying the equation:

⃗AC^2 = 4 + 9 - 6

⃗AC^2 = 7

Taking the square root of both sides:

⃗AC = √7

Since ⃗BD = ⃗AC, ⃗BD = √7.

Now, let's calculate ⃗AC ⋅ ⃗BD using the equation ⃗AC ⋅ ⃗BD = ⃗AC^2 cos(θ). Since ∠B = 60°, we can substitute cos(60°) = 0.5:

⃗AC ⋅ ⃗BD = √7^2 * 0.5

Simplifying the equation:

⃗AC ⋅ ⃗BD = 7 * 0.5

⃗AC ⋅ ⃗BD = 3.5

Therefore, ⃗AC ⋅ ⃗BD = 3.5.

Diagram

To create a diagram indicating the vectors, we can draw a trapezoid ABCD with angles ∠A = 60° and ∠B = 60°. We can label the vectors ⃗AB = 2 and ⃗CD = 3. The vector ⃗AC can be represented as a diagonal connecting points A and C, and the vector ⃗BD can be represented as a diagonal connecting points B and D.