Step-by-step Solution:

1. Understand the vector representation of the triangle.

Given a triangle PQR with sides a = QR, b = RP, and c = PQ. The vectors representing the sides of a triangle taken in order sum to zero. Therefore, we have: $\vec{PQ} + \vec{QR} + \vec{RP} = \vec{0}$ Substituting the given vector names: $c + a + b = 0$ This vector relationship is crucial. From this, we can express one vector in terms of the other two, for instance: $c = -a - b$

2. Simplify the given equation.

The given equation is: ${{a \cdot (c - b)} \over {c \cdot (a - b)}} = {{|a|} \over {|a| + |b|}}$ We will simplify the numerator and the denominator of the left-hand side using the relation c = -a - b.

Numerator: a . (c - b) Substitute c = -a - b: $a \cdot ((-a - b) - b) = a \cdot (-a - 2b)$ $= - (a \cdot a) - 2(a \cdot b)$ $= -|a|^2 - 2(a \cdot b)$

Denominator: c . (a - b) Substitute c = -a - b: $(-a - b) \cdot (a - b) = -(a + b) \cdot (a - b)$ Using the vector identity corresponding to the difference of squares $(x+y)(x-y) = x^2-y^2$: $= - (a \cdot a - b \cdot b) = -(|a|^2 - |b|^2) = |b|^2 - |a|^2$

3. Substitute the simplified expressions back into the equation.

Plugging the simplified numerator and denominator back into the original equation, we get: ${{ -|a|^2 - 2(a \cdot b) } \over { |b|^2 - |a|^2 }} = {{|a|} \over {|a| + |b|}}$

4. Substitute the given magnitudes.

We are given |a| = 3 and |b| = 4. Let's compute the necessary values:

• $|a|^2 = 3^2 = 9$
• $|b|^2 = 4^2 = 16$
• |a| + |b| = 3 + 4 = 7

Substituting these values into the equation from Step 3: ${{ -9 - 2(a \cdot b) } \over { 16 - 9 }} = {{3} \over {7}}$ ${{ -9 - 2(a \cdot b) } \over { 7 }} = {{3} \over {7}}$

5. Solve for the dot product a . b.

By comparing the numerators (since the denominators are equal and non-zero), we have: $-9 - 2(a \cdot b) = 3$ $-2(a \cdot b) = 3 + 9$ $-2(a \cdot b) = 12$ $a \cdot b = -6$

6. Calculate the value of $|a x b|^2$.

We use Lagrange's identity, which relates the cross product and dot product of two vectors: $|a \times b|^2 + (a \cdot b)^2 = |a|^2 |b|^2$ Rearranging the formula to solve for $|a x b|^2$: $|a \times b|^2 = |a|^2 |b|^2 - (a \cdot b)^2$ Now, substitute the known values:

• $|a|^2 = 9$
• $|b|^2 = 16$
• $(a \cdot b)^2 = (-6)^2 = 36$

$|a \times b|^2 = (9)(16) - 36$ $|a \times b|^2 = 144 - 36$ $|a \times b|^2 = 108$

Thus, the value of $|a x b|^2$ is 108.