Let for a triangle ABC, A B = - 2 i + j + 3 k C B = i + j + k C A = 4 i + 3 j + k If 0 and area of the triangle A B C is 5 6 , then C B · C A is equal to

Question

$Let for a triangle ABC,$

$\vec{A B} = - 2 \hat{i} + \hat{j} + 3 \hat{k}$

$\vec{C B} = α \hat{i} + β \hat{j} + γ \hat{k}$

$\vec{C A} = 4 \hat{i} + 3 \hat{j} + δ \hat{k}$

$If δ > 0 and area of the triangle A B C is 5 \sqrt{6}, then \vec{C B} \cdot \vec{C A} is equal to$

Smart Achievers · Accepted Answer

(1)

$Since, we know \vec{A B} + \vec{B C} + \vec{C A} = \vec{0}$

$\Rightarrow α = 2, β = 4, γ - δ = 3$

$Now, \frac{1}{2}$ $| \vec{A B} \times \vec{A C} |$ $= 5 \sqrt{6}$

${(δ - 9)}^{2} + {(2 δ + 12)}^{2} + 100 = 600$

$\Rightarrow δ = 5, γ = 8$

$Hence, \vec{C B} \cdot \vec{C A} = 60$

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