I am new to vector geometry. I have a set of two equations:-
$p_2 \cdot x = (p_3 - \bar{p}) \cdot b_1$
$p_2 \cdot y = (p_3 - \bar{p}) \cdot b_2$
where, $p_2(x,y)$ is a 2D vector, $p_3(x,y,z)$ is a 3D vector, $b_1$, $b_2$ are also 3D. Now given that I know $p_2$, $b_1$, $b_2$, $\bar{p}$. Can we find out $p_3$ using the above equations. If so how?
I bought the $b_1$ to the denominator as we do in scalar equations, but I don't really think that it works.
You essentially have two equations in three unknowns, for which there is not a unique solution. If $p_3=(x,y,z)$ then since $\bar{p}$ and $b_2$ are known, even if both expressions involving 2d vectors on the left sides compute to known numbers, you can only get so far as $$[1] \ \ p_3 \cdot b_1=k_1,\ \ p_3 \cdot b_2=k_2,$$ where $k_1=p_2.x+\bar{p}\cdot b_1$ and $k_2=p_2.y + \bar{p}\cdot b_2.$ It is unclear (to me) just what $p_2.x$ and $p_2.y$ mean, and whether $x,y$ denote two dimensional vectors which are known. But the point is, whatever they denote, if their values are known real numbers then $k_1,k_2$ are just two real numbers, and then replacing $p_3$ by its unknown coordinates $(x,y,z)$ and replacing $b_1,b_2$ by whatever numerical coordinates they have, the system [1] ends up as two linear equations in the three unknowns $x,y,z$. But two linear equations in three unknowns are never enough to determine the values of those unknowns.