Determine the value of a

Question

Please,this is the last problem that i need to solve:

Can you help me?I don't know how to start.

Determine the value of a such that the vectors A = (a, 4), B = (2,5) are parallel.

Answer 1

Two vectors $A$ and $B$ are parallel, when the following equality holds $$A=kB$$ for some $k\in\mathbb{R}$ Thus we have a system of equations $$ka=2$$ $$4k=5$$ $$\implies$$ $$k=\frac{5}{4}$$ Substituting this in the first equation we get $$\frac{5}{4}a=2$$ $$\implies$$ $$a=\frac{8}{5}$$

Thus $$A = \begin{pmatrix} \frac{8}{5} \\ 4 \end{pmatrix}$$

Answer 2

If $v $ is parallel to $u $, then there exists a number $\lambda$ to which we call a scalar such that $v = \lambda u $.

Because a vector is only equal to another if all components are equal, one by one, we are able to write a system of equations.

You can say your vectors are parallel when there is a $\lambda$ such that $(a, 4) = \lambda (2, 5) $. Equaling component by component we get

$ \begin {cases} a = 2\lambda\\ 4 = 5\lambda \end {cases}$