Question

Даны векторы АВ=(5;α;β) и AC=(2; -4;8). Если точки А, В и С лежат на одной прямой, то сумма α+β равна...

Answer 1

Равна...
a=-4/2=-2
b=8/2=4

Answer 2

Given Vectors

The given vectors are: - Vector AB = (5; α; β) - Vector AC = (2; -4; 8)

Condition for Points to Lie on the Same Line

For points A, B, and C to lie on the same line, the vectors AB and AC must be linearly dependent. This means that one of the vectors can be written as a scalar multiple of the other.

Finding α and β

To find α and β, we can use the fact that the vectors AB and AC are linearly dependent. This means that one of the vectors can be written as a scalar multiple of the other.

Let's find the values of α and β based on this condition.

Calculation

The vector AB can be written as a scalar multiple of the vector AC if and only if the following equation holds: AB = k * AC, where k is a scalar.

Using the given vectors: (5; α; β) = k * (2; -4; 8)

By comparing the corresponding components: 5 = 2k α = -4k β = 8k

Solving for k: k = 5 / 2 α = -4 * (5 / 2) = -10 β = 8 * (5 / 2) = 20

Sum of α and β

The sum of α and β is: α + β = -10 + 20 = 10