Solveeit Logo
Login

Question

Question: If \(\mathbf{\alpha} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k},\mathbf{\beta} = - \mathbf{i} + 2\math...

If α=2i+3j−k,β=−i+2j−4k\mathbf{\alpha} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k},\mathbf{\beta} = - \mathbf{i} + 2\mathbf{j} - 4\mathbf{k} and γ=i+j+k,\mathbf{\gamma} = \mathbf{i} + \mathbf{j} + \mathbf{k}, then

(α×β).(α×γ)(\mathbf{\alpha} \times \mathbf{\beta}).(\mathbf{\alpha} \times \mathbf{\gamma}) is equal to

A

60

B

64

C

74

D

– 74

Answer

– 74

Explanation

Solution

α.[γ×(α×β)]=α.[(γ.β)α−(γ.α)β]\mathbf{\alpha}.\lbrack\mathbf{\gamma} \times (\mathbf{\alpha} \times \mathbf{\beta})\rbrack = \mathbf{\alpha}.\lbrack(\mathbf{\gamma}.\mathbf{\beta})\mathbf{\alpha} - (\mathbf{\gamma}.\mathbf{\alpha})\mathbf{\beta}\rbrack

=(α)2(γ.β)−(γ.α)(β.α)=14(−3)−(4)(8)=−74= (\mathbf{\alpha})^{2}(\mathbf{\gamma}.\mathbf{\beta}) - (\mathbf{\gamma}.\mathbf{\alpha})(\mathbf{\beta}.\mathbf{\alpha}) = 14( - 3) - (4)(8) = - 74.