Let $$\vec{a} = \hat{i} + 2\hat{j} + \lambda\hat{k}$$, $$\vec{b} = 3\hat{i} - 5\hat{j} - \lambda\hat{k}$$, $$\vec{a} \cdot \vec{c} = 7$$, $$2(\vec{b} \cdot \vec{c}) + 43 = 0$$, $$\vec{a} \times \vec{c} = \vec{b} \times \vec{c}$$, then $$|\vec{a} \cdot \vec{b}|$$ is equal to

We need to find $$|\vec{a} \cdot \vec{b}|$$ given the conditions $$\vec{a} \cdot \vec{c} = 7$$, $$2(\vec{b} \cdot \vec{c}) + 43 = 0$$, and $$\vec{a} \times \vec{c} = \vec{b} \times \vec{c}$$.

$$\vec{a} \times \vec{c} = \vec{b} \times \vec{c} \implies (\vec{a} - \vec{b}) \times \vec{c} = \vec{0}$$

This means $$\vec{a} - \vec{b}$$ is parallel to $$\vec{c}$$, so $$\vec{c} = t(\vec{a} - \vec{b})$$ for some scalar $$t$$.

$$\vec{a} - \vec{b} = (\hat{i} + 2\hat{j} + \lambda\hat{k}) - (3\hat{i} - 5\hat{j} - \lambda\hat{k}) = -2\hat{i} + 7\hat{j} + 2\lambda\hat{k}$$

From $$\vec{a} \cdot \vec{c} = 7$$:

$$t[(-2)(1) + (7)(2) + (2\lambda)(\lambda)] = 7 \implies t(12 + 2\lambda^2) = 7 \quad \text{...(i)}$$

From $$2(\vec{b} \cdot \vec{c}) + 43 = 0 \implies \vec{b} \cdot \vec{c} = -\frac{43}{2}$$:

$$t[(-2)(3) + (7)(-5) + (2\lambda)(-\lambda)] = -\frac{43}{2} \implies t(-41 - 2\lambda^2) = -\frac{43}{2} \quad \text{...(ii)}$$

Dividing (i) by (ii):

$$\frac{12 + 2\lambda^2}{-(41 + 2\lambda^2)} = \frac{7}{-43/2} = -\frac{14}{43}$$

$$43(12 + 2\lambda^2) = 14(41 + 2\lambda^2)$$

$$516 + 86\lambda^2 = 574 + 28\lambda^2$$

$$58\lambda^2 = 58 \implies \lambda^2 = 1$$

$$\vec{a} \cdot \vec{b} = (1)(3) + (2)(-5) + (\lambda)(-\lambda) = 3 - 10 - \lambda^2 = 3 - 10 - 1 = -8$$

$$|\vec{a} \cdot \vec{b}| = 8$$

The correct answer is $$8$$.