Question

Find all constants a such that the vectors (a, 4) and (2, 5) are parallel.

Answer 1

Answer:

$\displaystyle a = \frac{8}{5}$.

Step-by-step explanation:

Two vectors $(x_1,\, y_1)$ and $(x_2,\, y_2)$ are parallel to one another if and only if the ratio between their corresponding components are equal:

$\displaystyle \frac{x_1}{x_2} = \frac{y_1}{y_2}$.

Equivalently:

$\displaystyle \frac{x_1}{y_1} = \frac{x_2}{y_2}$.

For the two vectors in this equation to be parallel to one another:

$\displaystyle \frac{a}{4} = \frac{2}{5}$.

Solve for $a$:

$\displaystyle a = \frac{8}{5}$.

$\displaystyle \frac{8}{5}$ would be the only valid value of $a$; no other value would satisfy the $\displaystyle \frac{x_1}{y_1} = \frac{x_2}{y_2}$ equation.