The vectors 60\hat{i}+3\hat{j}, 40\hat{i}-8\hat{j} and \beta\hat{i}-52\hat{j} are collinear if:

Explanation

Let the points be A = 60\hat{i}+3\hat{j}, B = 40\hat{i}-8\hat{j}, and C = \beta\hat{i}-52\hat{j}. For them to be collinear, the vectors \vec{AB} and \vec{BC} must be proportional. \vec{AB} = (40-60)\hat{i} + (-8-3)\hat{j} = -20\hat{i} - 11\hat{j}. \vec{BC} = (\beta-40)\hat{i} + (-52-(-8))\hat{j} = (\beta-40)\hat{i} - 44\hat{j}. The ratio of their components must be equal: \frac{-20}{\beta-40} = \frac{-11}{-44}. Simplifying gives \frac{-20}{\beta-40} = \frac{1}{4} \implies \beta - 40 = -80 \implies \beta = -40.

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