express vector with other vectors

Question

I have 4 vectors, A(2,2,2), P(2,3,4), Q(3,−1,0), R(-4,-1,-3). I found that they are linearly independent. The next question is to express A vector with other vectors.

Answer 1

First, as MRK correctly points out, these are linearly dependent.

Hint

If you want to write $\vec{A}$ in terms of others, you need some numbers $p,q,r$ such that $$ \vec{A} = p\vec{P} + q\vec{Q} + r\vec{R} = \begin{bmatrix} \vec{P} & \vec{Q} & \vec{R} \end{bmatrix} \begin{pmatrix} p\\q\\r \end{pmatrix}, $$ which amounts to solving a $3 \times 3$ system. Can you finish it?

Answer 2

The four vectors must be linearly dependent because they live in a three-dimensional vector space.

What I'm guessing you need is to find scalars $a,b,c$ such that

$$aP + bQ + cR = A.$$

Or,

$$a\left[\begin{matrix} 2 \\ 3 \\ 4\end{matrix}\right] + b\left[\begin{matrix} 3 \\ -1 \\ 0\end{matrix}\right] + c\left[\begin{matrix} -4 \\ -1 \\ -3\end{matrix}\right] = \left[\begin{matrix} 2 \\ 2 \\ 2\end{matrix}\right].$$

This means solving the system:

$$2a + 3b - 4c = 2 \\ 3a - b = 2 \\ 4a - 3c = 2.$$

Can you take it from there?