Question

$\text{[gex36]}$ Vector divisions? Considering the two kinds of multiplications of vectors \text{---} the dot product and the cross product \text{---} are there meaningful ways two divide by a vector? One way to put that question is by asking to find the vector $X$ that solves the two equations, $A \cdot X = c$, $A \times X = C$, for a given scalar $c$ and a given vectors $C$ and $A$. Show that a solution only exists if $A \perp C$ and that the solution then reads $X = \frac{C \times A + cA}{A \cdot A}$. The quotient theorem in tensor analysis [gmd5] is an immensely useful elaboration of the concept of vector division.

          $\text{[gex36]}$ Vector divisions?
Considering the two kinds of multiplications of vectors \text{---} the dot product and the cross product \text{---} are there meaningful ways two divide by a vector? One way to put that question is by asking to find the vector $X$ that solves the two equations,
$A \cdot X = c$, $A \times X = C$,
for a given scalar $c$ and a given vectors $C$ and $A$. Show that a solution only exists if $A \perp C$ and that the solution then reads
$X = \frac{C \times A + cA}{A \cdot A}$.
The quotient theorem in tensor analysis [gmd5] is an immensely useful elaboration of the concept of vector division.

[gex36] Vector divisions?
Considering the two kinds of multiplications of vectors — the dot product and the cross product — are there meaningful ways two divide by a vector? One way to put that question is by asking to find the vector X that solves the two equations,
A · X = c, A × X = C,
for a given scalar c and a given vectors C and A. Show that a solution only exists if A ⊥ C and that the solution then reads
X = (C × A + cA)/(A · A).
The quotient theorem in tensor analysis [gmd5] is an immensely useful elaboration of the concept of vector division.

Added by Jose Angel S.

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