Question

Даны точки E,F и K, причем вектор EF(1;-2;3) и EK(-2;4;6). Лежат ли эти точки на одной прямой? помоогите пожалуйста.. очень очень прошу

Answer 1

точки E,F и K лежат на одной пряммой если векторы EF(1;-2;3) и EK(-2;4;6) коллинеарны, т.е. если справедливо соотношение EK=k*EF, где k- некоторое действительное число, т.е.

справедливы соотношения

$\frac{-2}{1}=\frac{4}{-2}=\frac{6}{3}=k;$

или

-2=-2=2=k, что очевидно неверно (-2 не равно 2)

значит данные точки на одной пряммой не лежат

Answer 2

Problem Analysis

We are given three points E, F, and K, and the vectors EF(1,-2,3) and EK(-2,4,6). We need to determine if these points lie on the same line.

Solution

To check if the points E, F, and K lie on the same line, we can use the fact that if two vectors are parallel, then the points they represent lie on the same line.

Let's calculate the vector FK by subtracting the vector EK from the vector EF:

FK = EF - EK = (1, -2, 3) - (-2, 4, 6) = (1 + 2, -2 - 4, 3 - 6) = (3, -6, -3)

Now, we can check if the vectors EF and FK are parallel. If they are parallel, then the points E, F, and K lie on the same line.

To check if two vectors are parallel, we can compare their direction ratios. If the ratios of the corresponding components are equal, then the vectors are parallel.

Let's compare the direction ratios of EF and FK:

Direction ratios of EF: (1, -2, 3)

Direction ratios of FK: (3, -6, -3)

Comparing the direction ratios, we can see that the ratios are equal:

1/3 = -2/-6 = 3/-3

Since the direction ratios of EF and FK are equal, we can conclude that the vectors EF and FK are parallel. Therefore, the points E, F, and K lie on the same line.

Conclusion

The points E, F, and K lie on the same line.

Note: The solution provided above is based on the given information and calculations. Please double-check the calculations and verify the solution.