Is there a concise way to express $ \left(\vec{x}\cdot\vec{z}\right)\left(\vec{y}\cdot\vec{z}\right) $

Question

Given that $\vec{x}$, $\vec{y}$ & $\vec{z}$ are three arbitrary vectors in $\mathbb{R}^3$, is there a concise way to express $ \left(\vec{x}\cdot\vec{z}\right)\left(\vec{y}\cdot\vec{z}\right) $ in terms of $\vec{z}$, $\left(\vec{x}\times\vec{y}\right)$ and maybe $\left(\vec{x}\cdot\vec{y}\right)$ only?

Answer 1

$(x\times y)\times z=(y\cdot z)x-(x\cdot z)y\\ x\times(y\times z)=(x\cdot y)z-(x\cdot z)y$

Subtracting these we get $(y\cdot z)x-(x\cdot y)z$, and scalar multiplying by $z$, as $z\perp (x\times y)\times z$, it yields $$(y\cdot z)(x\cdot z) = -\ \left( (x\times (y\times z))\,\cdot z\right) \ + (x\cdot y)\,||z||^2$$

Well, this is not really nice. Probably with matrix multiplication, it is more useful, as $(x\cdot y)=x^Ty$ where vectors are regarded as culomn matrices. $$(x\cdot z)(y\cdot z)=x^Tzy^Tz \,.$$

Answer 2

$$ (y\cdot z)(x\cdot z) = (x\cdot y)\,||z||^2 -\ \left( (x\times z) \cdot (y\times z) \right) $$ This is obtained by expansion of the third term.