A line AB in three-dimensional space makes angles 45^ and 120^ with the positive x -axis and the positive y -axis respectively. If AB makes an acute a - Mathematics MCQ AIEEE 2026 | ExamTest.live

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A line A B in three-dimensional space makes angles 4 5^{\circ} and 12 0^{\circ} with the positive x -axis and the positive y -axis respectively. If A B makes an acute angle θ with the positive z -axis, then θ equals:

Solution & Step-by-step Explanation

Let l, m, n be direction cosines. l = cos 4 5^{\circ} = 1/ 2 . m = cos 12 0^{\circ} = - 1/2 . n = cos θ .We know l^{2} + m^{2} + n^{2} = 1 . (1/ 2)^{2} + (- 1/2)^{2} + cos^{2} θ = 1 1/2 + 1/4 + cos^{2} θ = 1 3/4 + cos^{2} θ = 1 ⟹ cos^{2} θ = 1/4 . cos θ = \pm 1/2 . Since θ is acute, cos θ = 1/2 . θ = 6 0^{\circ} .

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A line A B in three-dimensional space makes angles 4 5^{\circ} and 12 0^{\circ} with the positive x -axis and the positive y -axis respectively. If A B makes an acute angle θ with the positive z -axis, then θ equals:

4 5^{\circ}

6 0^{\circ}

7 5^{\circ}

3 0^{\circ}

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