maths vectors

the vectors 2a + kb and 5a + 3b are parallel. find the value of k.

What must you multiply 5 by to get 2?

So multiply 3 by the same thing to get k.

All this is coming from simply comparing the coefficients of vectors a and b.

so u just multiply 5a + 3b by 2/5?
could you explain it to me pls?

If the vector $2\mathbf{a} + k \mathbf{b}$ is parallel to $5\mathbf{a} + 3\mathbf{b}$, then this second vector is a scalar multiple of the first one.

So, we have that $2\mathbf{a} + k \mathbf{b} = \lambda(5\mathbf{a} + 3\mathbf{b})$ where $\lambda$ is the scalar.

We can rearrange this into $(2- 5\lambda)\mathbf{a} + (k - 3\lambda)\mathbf{b} = \mathbf{0}$, and because the two vectors $\mathbf{a},\mathbf{b}$ are not scalar multiples of another, we must have that the coefficients of these vectors are zero in order to have the zero vector on the RHS.

This implies that

$\begin{cases} 2 - 5\lambda = 0 \\ k - 3\lambda = 0 \end{cases}$

and from the first equation we find that this scalar multiple is precisely $\lambda = \dfrac{2}{5}$ which we can substitute into the second equation and that that $k = 3 \cdot \dfrac{2}{5}$.



That's the expalanation, and the quick way is to find the scalar $\lambda$ by simply observing what you need to multiple the coefficient of $\mathbf{a}$ by in the second vector in order to get the coefficient of $\mathbf{a}$ in the first vector. Then use it on the $\mathbf{b}$ vectors.