Question

СРОЧНО НУЖНО СРОЧНО! ДАЮ 20 БАЛЛ Учитывая точки A (2; -1), B (-4; 3), C (5; -1), D (1: a), при каком значении вектора a) AB и CD колинниары? б) При каком значении вектора AB и CD перпендикулярны?

Answer 1

Ответ:

AB=(-4-2,3-(-1))=(-6,4)

CD=(1-5,a+1)=(-4,a+1)

-6/-4=4/(a+1)

6(a+1)=16

3a+3=8

3a=5

a=5/3

9a=9*(5/3)=15

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Answer 2

Finding the Condition for Collinearity of Vectors AB and CD

To determine the condition for vectors AB and CD to be collinear, we need to find the relationship between their components.

Let's start by finding the vectors AB and CD using the given points A(2, -1), B(-4, 3), C(5, -1), and D(1, a).

The vector AB can be obtained by subtracting the coordinates of point A from the coordinates of point B:

AB = (xB - xA, yB - yA) = (-4 - 2, 3 - (-1)) = (-6, 4).

Similarly, the vector CD can be obtained by subtracting the coordinates of point C from the coordinates of point D:

CD = (xD - xC, yD - yC) = (1 - 5, a - (-1)) = (-4, a + 1).

Now, for vectors AB and CD to be collinear, they must be scalar multiples of each other. In other words, one vector must be a scalar multiple of the other.

Let's assume that AB and CD are collinear. This means that there exists a scalar k such that:

AB = k * CD.

Comparing the components of the two vectors, we get the following equations:

-6 = k * (-4) (for the x-component) 4 = k * (a + 1) (for the y-component).

To find the value of a for which AB and CD are collinear, we can solve these equations simultaneously.

From the first equation, we have:

k = -6 / (-4) = 3/2.

Substituting this value of k into the second equation, we get:

4 = (3/2) * (a + 1).

Simplifying the equation, we have:

4 = (3/2) * a + 3/2.

Multiplying both sides of the equation by 2 to eliminate the fraction, we get:

8 = 3a + 3.

Subtracting 3 from both sides of the equation, we have:

5 = 3a.

Dividing both sides of the equation by 3, we get:

a = 5/3.

Therefore, for vectors AB and CD to be collinear, the value of a must be 5/3.

Finding the Condition for Perpendicularity of Vectors AB and CD

To determine the condition for vectors AB and CD to be perpendicular, we need to find the relationship between their dot product.

Two vectors are perpendicular if and only if their dot product is zero.

Let's calculate the dot product of vectors AB and CD:

AB · CD = (xAB * xCD) + (yAB * yCD).

Substituting the values of the components, we have:

AB · CD = (-6 * -4) + (4 * (a + 1)).

Simplifying the equation, we get:

AB · CD = 24 + 4(a + 1).

To find the value of a for which AB and CD are perpendicular, we need to solve the equation:

24 + 4(a + 1) = 0.

Expanding and simplifying the equation, we have:

24 + 4a + 4 = 0.

Combining like terms, we get:

4a + 28 = 0.

Subtracting 28 from both sides of the equation, we have:

4a = -28.

Dividing both sides of the equation by 4, we get:

a = -7.

Therefore, for vectors AB and CD to be perpendicular, the value of a must be -7.

In summary: - Vectors AB and CD are collinear when a = 5/3. - Vectors AB and CD are perpendicular when a = -7.