How to solve vector equation?

Given the vector relation $(mathbf{a}+3 mathbf{b}) perp(7 mathbf{a}-5 mathbf{b})$, $(mathbf{a}-4 mathbf{b}) perp(7 mathbf{a}-2 mathbf{b})$, find the angle between $mathbf{a}$ and $mathbf{b}$.

$$begin{array}{c}
text{Solving process: } qquad because(mathbf{a}+3 mathbf{b}) cdot(7 mathbf{a}-5mathbf{b}) = 0,(mathbf{a}-4 mathbf{b}) cdot(7 mathbf{a}-2 mathbf{b}) = 0
therefore quad 7 mathbf{a}^{2}+16 mathbf{a} cdot mathbf{b}-15 mathbf{b}^{2} = 0,7 mathbf{a}^{2}-30 mathbf{a} cdot mathbf{b}+8 mathbf{b}^{2} = 0
quad mathbf{a}^{2} = mathbf{b}^{2} = 2 mathbf{a} cdot mathbf{b}
Rightarrow cos langle mathbf{a}, mathbf{b}rangle = frac{mathbf{a} cdot mathbf{b}}{mathbf{a}|cdot mathbf{b}|} = frac{1}{2}
end{array}$$

a = {x1, y1, z1};
b = {x2, y2, z2};
Solve[{7 a . a + 16 a . b - 15 b . b == 0, 
   7 a . a - 30 a . b + 8 b . b == 0, 
   cosa == a . b/(Norm[a] Norm[b])}, 
  cosa, {x1, y1, z1, x2, y2, z2}] // FullSimplify

How to use Mathematica to directly solve the above vector equation and find the angle between vector $mathbf{a}$ and vector $mathbf{b}$? I wonder if there is any other easier way.

With[{a = {a1, a2, a3}, b = {b1, b2, b3}}, 
     Eliminate[{(a + 3 b) . (7 a - 5 b) == 0, (a - 4 b) . (7 a - 2 b) == 0,
                cang == a . b/(Sqrt[a . a] Sqrt[b . b])}, Join[a, b]]]
   4 cang^2 == 1

Solve[%, cang]
   {{cang -> -1/2}, {cang -> 1/2}}

{(a + 3 b) . (7 a - 5 b) == 0, (a - 4 b) . (7 a - 2 b) == 0} /.
{a -> {1, 0, 0}, b -> Append[AngleVector[2 π/3], 0]}
   {False, False}

{(a + 3 b) . (7 a - 5 b) == 0, (a - 4 b) . (7 a - 2 b) == 0} /.
{a -> {1, 0, 0}, b -> Append[AngleVector[π/3], 0]}
   {True, True}