Given vectors AB = (5; α; β) and AC = (2; -4; 8). If points A, B and C lie on one straight line

Given vectors AB = (5; α; β) and AC = (2; -4; 8). If points A, B and C lie on one straight line, then the sum α + β is equal to …

Let vectors AB = (5; α; β) and AC = (2; – 4; 8) be given. If points A, B and C lie on one straight line, then the vectors AB and AC will be collinear, and then their corresponding coordinates will be proportional. Let us find the proportionality coefficient k from the fact that the vector AB has the abscissa x₁ = 5, and the vector AC has the abscissa x₂ = 2. We obtain that k = x₁ / x₂ = 5/2 = 2.5. Hence, y₁ / y₂ = k and z₁ / z₂ = k. It is known from the condition of the problem that у₁ = α; y₂ = – 4; z₁ = β; z₂ = 8. Substitute the values ​​of the known quantities into the equalities and find the values ​​of the ordinate α and the applicates β of the vector AB:
α / (- 4) = 2.5;
α = 2.5 ∙ (- 4);
α = – 10;
β / 8 = 2.5;
β = 2.5 ∙ 8;
β = 20;
then the sum of the values ​​of the ordinate and the applicate will be:
α + β = – 10 + 20;
α + β = 10.
Answer: the sum of the values ​​of the ordinate and the applicate of the vector AB will be equal to 10.