Let a = 2i + α j + k, b = -i + k, c = β j - k, where α and β are integers and αβ = -6. Let the values of the o…

Question

Let **$$\vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}$$**, **$$\vec{b} = -\hat{i} + \hat{k}$$**, **$$\vec{c} = \beta \hat{j} - \hat{k}$$**, where **$$\alpha$$** and **$$\beta$$** are integers and **$$\alpha\beta = -6$$**. Let the values of the ordered pair **$$(\alpha, \beta)$$** for which the area of the parallelogram of diagonals **$$\vec{a} + \vec{b}$$** and **$$\vec{b} + \vec{c}$$** is **$$\frac{\sqrt{21}}{2}$$**, be **$$(\alpha_1, \beta_1)$$** and **$$(\alpha_2, \beta_2)$$**. Then **$$\alpha_1^2 + \beta_1^2 - \alpha_2\beta_2$$** is equal to:

Accepted Answer

D. $$19$$

Answer

A. $$17$$

Answer

B. $$24$$

Answer

C. $$21$$

Answer & step-by-step solution

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