How to find one vector of $w$ in the Euclidean inner product inside a vector space

Question

if $\langle~~,~\rangle$ symbolizes the Euclidean inner product inside the vector space $\Bbb R^3$, find one vector of w for which

$\langle w,v_1 \rangle + \langle w,v_2 \rangle = 2 \langle w,v_3 \rangle$

where $v_1= [1,0,1], v_2 = [1,2,1], v_3 = [1,3,10]$

How can I solve this?

Answer 1

In other words, you want $w$ to be orthogonal to $v_1+v_2-2v_3$. Just take $w$ to be the cross product of this vector with any other not parallel to it.

Answer 2

Let the unknown vector $w = [x, y, z]$ where $(x, y, z)$ are variables. Now, plug it into the given constraints: \begin{align*} &\langle w, v_1\rangle + \langle w, v2\rangle = 2 \langle w, v_3 \rangle \\ &\langle [x, y, z], [1, 0, 1]\rangle + \langle [x, y, z], [1, 2, 1]\rangle = 2 \langle [x, y, z], [1, 3, 10] \rangle \\ &(x +z) + (x + 2y + z) = 2(x+3y+10z) \\ &2x + 2y+2z = 2x + 6y + 20z \\ &2y + 2z = 6y + 20z \\ &-4y = 18z \\ &y = \frac{-18z}{4} \end{align*}

So,we have no constraints on $x$, and we know that $y = \frac{-18z}{4}$. Therefore, $w = [x, -18z/4, z], \forall x, z \in \mathbb R$