Understanding Direction Ratios in 3D Geometry

Direction ratios are numbers that are proportional to the direction cosines of a line. They define the direction of a line in three-dimensional space. If a line has direction cosines \(\ell, m, n\), then its direction ratios can be any set of numbers $a, b, c$ such that \(a = k\ell\), $b = km$, $c = kn$ for some non-zero scalar $k$. In other words, the ratio \(a:\ell = b:m = c:n\) holds.

Analyzing Statement 1: Direction Ratios of y-axis

Statement 1 says: The direction ratios of y-axis can be <0, 4, 0>.

• The y-axis is a line that goes infinitely in the positive and negative y direction.
• Its direction cosines are <0, 1, 0>. This means the angle it makes with the x-axis is 90 degrees (cosine is 0), with the y-axis is 0 degrees (cosine is 1), and with the z-axis is 90 degrees (cosine is 0).
• Direction ratios of the y-axis can be obtained by multiplying its direction cosines by any non-zero scalar $k$. So, possible direction ratios are <\(k \times 0\), \(k \times 1\), \(k \times 0\)> = <0, $k$, 0>, where \(k \neq 0\).
• The given set of direction ratios is <0, 4, 0>.
• Comparing <0, 4, 0> with <0, $k$, 0>, we can see that if we take $k=4$, we get <0, 4, 0>. Since $k=4$ is a non-zero scalar, <0, 4, 0> is indeed a valid set of direction ratios for the y-axis.

Therefore, Statement 1 is correct.

Analyzing Statement 2: Direction Ratios of a Line Perpendicular to z-axis

Statement 2 says: The direction ratios of a line perpendicular to z-axis can be <5, 6, 0>.

• A line is perpendicular to the z-axis if its direction vector is orthogonal to the direction vector of the z-axis.
• The z-axis has direction ratios <0, 0, 1> (or any multiple like <0, 0, $k$>, \(k \neq 0\)).
• Let the direction ratios of the line perpendicular to the z-axis be <a, b, c>.
• For two lines with direction ratios <\(a_1\), \(b_1\), \(c_1\)> and <\(a_2\), \(b_2\), \(c_2\)> to be perpendicular, the sum of the product of their corresponding direction ratios must be zero: \(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\).
• Here, the first line is perpendicular to the z-axis. Let's use the direction ratios of the z-axis as <0, 0, 1> and the direction ratios of the perpendicular line as <a, b, c>.
• The condition for perpendicularity is $a(0) + b(0) + c(1) = 0$.
• This simplifies to $c = 0$.
• So, any line perpendicular to the z-axis must have direction ratios of the form <a, b, 0>, where at least one of 'a' or 'b' is non-zero.
• The given set of direction ratios is <5, 6, 0>.
• Comparing <5, 6, 0> with <a, b, 0>, we see that it fits the form where $a=5$, $b=6$, and $c=0$. Since $a=5$ and $b=6$ are not both zero, this is a valid set of direction ratios for a line perpendicular to the z-axis.

Therefore, Statement 2 is correct.

Conclusion on the Statements

Both Statement 1 regarding the direction ratios of the y-axis and Statement 2 regarding the direction ratios of a line perpendicular to the z-axis are found to be correct based on the properties of direction ratios and perpendicular lines in 3D geometry.

Based on the analysis, both statements are correct.

Revision Table: Key Concepts

Additional Information: Geometric Interpretation

The direction ratios <a, b, c> can be thought of as the components of a vector parallel to the line. For instance, the vector \(\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}\) points in the same direction as the line. The direction cosines are the components of the unit vector in that direction: \(\hat{u} = \frac{a}{\sqrt{a^2+b^2+c^2}}\hat{i} + \frac{b}{\sqrt{a^2+b^2+c^2}}\hat{j} + \frac{c}{\sqrt{a^2+b^2+c^2}}\hat{k}\). The direction cosines are \(\ell = \frac{a}{\sqrt{a^2+b^2+c^2}}\), \(m = \frac{b}{\sqrt{a^2+b^2+c^2}}\), \(n = \frac{c}{\sqrt{a^2+b^2+c^2}}\).

For Statement 1, <0, 4, 0> as direction ratios means a vector \(0\hat{i} + 4\hat{j} + 0\hat{k} = 4\hat{j}\). This vector is clearly parallel to the y-axis.

For Statement 2, <5, 6, 0> as direction ratios means a vector \(5\hat{i} + 6\hat{j} + 0\hat{k}\). The z-axis is in the direction of the vector \(\hat{k}\) (or <0, 0, 1>). The dot product of \((5\hat{i} + 6\hat{j} + 0\hat{k})\) and \((0\hat{i} + 0\hat{j} + 1\hat{k})\) is $(5)(0) + (6)(0) + (0)(1) = 0$. A zero dot product indicates the vectors (and thus the lines they represent) are perpendicular.