Understanding Direction Cosines and Line Orientation

The direction cosines of a straight line in three-dimensional space are the cosines of the angles that the line makes with the positive x, y, and z axes. Let the angles be \(\alpha\), \(\beta\), and \(\gamma\) respectively, measured from the positive x, y, and z axes to the line. The direction cosines are denoted by \(\ell = \cos \alpha\), \(m = \cos \beta\), and \(n = \cos \gamma\). A fundamental property of direction cosines is that the sum of their squares is always equal to 1, i.e., \(\ell^2 + m^2 + n^2 = 1\).

In this question, we are given the direction cosines of a straight line as \(\left\langle {0,\;1,\;0} \right\rangle\). This means:

• \(\cos \alpha = 0\)
• \(\cos \beta = 1\)
• \(\cos \gamma = 0\)

We can determine the angles the line makes with the axes by finding the inverse cosine of these values:

• For the x-axis: \(\cos \alpha = 0\). The angle \(\alpha\) for which \(\cos \alpha = 0\) is \(90^\circ\). So, the line makes a \(90^\circ\) angle with the x-axis.
• For the y-axis: \(\cos \beta = 1\). The angle \(\beta\) for which \(\cos \beta = 1\) is \(0^\circ\). So, the line makes a \(0^\circ\) angle with the y-axis.
• For the z-axis: \(\cos \gamma = 0\). The angle \(\gamma\) for which \(\cos \gamma = 0\) is \(90^\circ\). So, the line makes a \(90^\circ\) angle with the z-axis.

A line that makes an angle of \(0^\circ\) with an axis is parallel to that axis. Similarly, a line that makes an angle of \(90^\circ\) with an axis is perpendicular to that axis. Since the line makes a \(0^\circ\) angle with the y-axis and \(90^\circ\) angles with both the x-axis and the z-axis, the line is parallel to the y-axis.

Analyzing the Options

Let's examine the given options based on our understanding of the line's orientation:

• Option 1: Parallel to x-axis

  A line parallel to the x-axis makes angles \(0^\circ\), \(90^\circ\), \(90^\circ\) with the x, y, and z axes respectively. Its direction cosines would be \(\cos 0^\circ = 1\), \(\cos 90^\circ = 0\), \(\cos 90^\circ = 0\), i.e., \(\left\langle {1,\;0,\;0} \right\rangle\). This does not match the given direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\).

• Option 2: Parallel to y-axis

  A line parallel to the y-axis makes angles \(90^\circ\), \(0^\circ\), \(90^\circ\) with the x, y, and z axes respectively. Its direction cosines would be \(\cos 90^\circ = 0\), \(\cos 0^\circ = 1\), \(\cos 90^\circ = 0\), i.e., \(\left\langle {0,\;1,\;0} \right\rangle\). This exactly matches the given direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\).

• Option 3: Parallel to z-axis

  A line parallel to the z-axis makes angles \(90^\circ\), \(90^\circ\), \(0^\circ\) with the x, y, and z axes respectively. Its direction cosines would be \(\cos 90^\circ = 0\), \(\cos 90^\circ = 0\), \(\cos 0^\circ = 1\), i.e., \(\left\langle {0,\;0,\;1} \right\rangle\). This does not match the given direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\).

• Option 4: Equally inclined to all the axes

  If a line is equally inclined to all axes, the angles it makes with the x, y, and z axes are equal, i.e., \(\alpha = \beta = \gamma\). Consequently, their cosines (the direction cosines) must also be equal, \(\cos \alpha = \cos \beta = \cos \gamma\). Let this common value be \(c\). The sum of squares of direction cosines must be 1, so \(c^2 + c^2 + c^2 = 1\), which means \(3c^2 = 1\), or \(c^2 = 1/3\). Thus, \(c = \pm 1/\sqrt{3}\). The direction cosines would be \(\left\langle {\pm \frac{1}{\sqrt{3}},\; \pm \frac{1}{\sqrt{3}},\; \pm \frac{1}{\sqrt{3}}} \right\rangle\). This does not match the given direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\).

Based on this analysis, the line with direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\) is parallel to the y-axis.

Additional Information: Direction Ratios

Related to direction cosines are direction ratios. Direction ratios of a line are any set of three numbers \(a, b, c\) that are proportional to the direction cosines \(\ell, m, n\). This means \(\frac{\ell}{a} = \frac{m}{b} = \frac{n}{c} = k\) for some constant \(k\). If \(a, b, c\) are direction ratios, then the direction cosines can be found using the formulas:

• \(\ell = \frac{a}{\pm\sqrt{a^2 + b^2 + c^2}}\)
• \(m = \frac{b}{\pm\sqrt{a^2 + b^2 + c^2}}\)
• \(n = \frac{c}{\pm\sqrt{a^2 + b^2 + c^2}}\)

The sign depends on the direction chosen for the line. For the direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\), one set of direction ratios could be \(\left\langle {0,\;1,\;0} \right\rangle\) itself, or any multiple like \(\left\langle {0,\;5,\;0} \right\rangle\) or \(\left\langle {0,\;-2,\;0} \right\rangle\). Direction ratios are useful because they don't have the constraint that the sum of their squares must be 1, making them sometimes easier to find initially.