A line makes angles of measure 45° and 60° with the positive directions of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis.

Q.7 A line makes angles of measure 45° and 60° with the positive directions of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis.

Doubtly Asked 28 Oct 2026 Board exam · March 2025 hsc 12th Save WhatsApp Report

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Answer: 60° or 120°

Let the line make angle \(\alpha\) with the X-axis; \(\beta = 45^\circ\), \(\gamma = 60^\circ\).

\(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \Rightarrow \cos^2\alpha + \frac12 + \frac14 = 1 \Rightarrow \cos^2\alpha = \frac14\).

\(\cos\alpha = \pm\frac12\), so \(\alpha = 60^\circ\) or \(\alpha = 120^\circ\).

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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A line makes angles of measure 45° and 60° with the positive directions of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis.

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