A unit vector a makes angle π/3 with i , π/4 with j and an acute angle θ with k .I. The value of θ is π/3 .II. The components of a are (1/2, 1/2, 1/2) - Mathematics MCQ Mathematics Mock Test - 9 2026 | ExamTest.live

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A unit vector a makes angle π /3 with \hat{i}, π /4 with \hat{j} and an acute angle θ with \hat{k} .I. The value of θ is π /3 .II. The components of a are (1/2, 1/2, 1/2) .Which of the above statement(s) is/are correct?

Solution & Step-by-step Explanation

Let direction cosines be l, m, n . l = cos (π /3) = 1/2 m = cos (π /4) = 1/ 2 n = cos θ Since it is a unit vector, l^{2} + m^{2} + n^{2} = 1 . (1/2)^{2} + (1/ 2)^{2} + cos^{2} θ = 1 1/4 + 1/2 + cos^{2} θ = 1 ⟹ 3/4 + cos^{2} θ = 1 ⟹ cos^{2} θ = 1/4 . cos θ = 1/2 (as θ is acute). θ = π /3 . Statement I is correct.Components are (l, m, n) = (1/2, 1/ 2, 1/2) .Statement II says (1/2, 1/2, 1/2), which is incorrect.

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A unit vector a makes angle π /3 with \hat{i}, π /4 with \hat{j} and an acute angle θ with \hat{k} .I. The value of θ is π /3 .II. The components of a are (1/2, 1/2, 1/2) .Which of the above statement(s) is/are correct?

Only I

Only II

Both I and II

Neither I nor II

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