A line makes angles \(\alpha\), \(\beta\) and \(\gamma\) with the positive directions of \(x\)-axis, \(y\)-axis and \(z\)-axis respectively such that \(\alpha + \beta = 90°\). Which of the following statements is/are correct? I. The maximum value of \(\cos\alpha + \cos\beta\) is \(\sqrt{2}\). II. The minimum value of \(\cos\alpha + \cos\beta + \cos\gamma\) is \(1\). Select the answer using the code given below.
I only
For a line in three-dimensional space, the direction cosines satisfy:
\(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\)
Given \(\alpha+\beta=90^\circ\), we have \(\beta=90^\circ-\alpha\), so:
\(\cos\beta=\sin\alpha\)
Therefore,
\(\cos^2\alpha+\cos^2\beta=\cos^2\alpha+\sin^2\alpha=1\)
Hence, \(\cos^2\gamma=0\), so \(\cos\gamma=0\).
Statement I: \(\cos\alpha+\cos\beta=\cos\alpha+\sin\alpha\). Its maximum value occurs at \(\alpha=45^\circ\):
\(\cos45^\circ+\sin45^\circ=\dfrac{1}{\sqrt2}+\dfrac{1}{\sqrt2}=\sqrt2\)
Hence, Statement I is correct.
Statement II: Since \(\cos\gamma=0\),
\(\cos\alpha+\cos\beta+\cos\gamma=\cos\alpha+\sin\alpha\)
For strictly acute values \(0^\circ<\alpha<90^\circ\), we have \(\cos\alpha+\sin\alpha>1\). The value approaches 1 only when \(\alpha\) approaches \(0^\circ\) or \(90^\circ\); therefore, under this interpretation, 1 is not attained as the minimum value.
Hence, Statement II is incorrect.
Correct Answer: I only
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Important Questions from Direction ratios and Direction cosines
A point on a line has coordinates (p + 1, p - 3, √2p) where p is any real number. What are the direction cosines of the line?
If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are the coordinates of the points on L whose distance from (1, -1, 1) is 2 units?
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A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
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