Analyzing Statements on Direction Ratios and Direction Cosines

The question asks us to evaluate two statements regarding the properties of lines in space, specifically focusing on direction ratios and direction cosines.

Statement 1: A line in space can have infinitely many direction ratios.

Let's understand what direction ratios and direction cosines are for a line in 3D space.

• Direction Cosines ($l, m, n$): These are the cosines of the angles that the line makes with the positive x, y, and z axes, respectively. A fundamental property of direction cosines is that the sum of their squares is always equal to 1, i.e., $\textrm{l}^2 + \textrm{m}^2 + \textrm{n}^2 = 1$.
• Direction Ratios ($a, b, c$): These are any set of three numbers that are proportional to the direction cosines $(l, m, n)$. If $(l, m, n)$ are the direction cosines, then $(a, b, c)$ are direction ratios if $\frac{a}{l} = \frac{b}{m} = \frac{c}{n} = \textrm{k}$ for some non-zero constant $\textrm{k}$. This implies $a = \textrm{k}l$, $b = \textrm{k}m$, $c = \textrm{k}n$.

Given a set of direction cosines $(l, m, n)$, any set of numbers $(\textrm{k}l, \textrm{k}m, \textrm{k}n)$ where $\textrm{k}$ is any non-zero real number will be a set of direction ratios for the same line. Since there are infinitely many choices for the non-zero value of $\textrm{k}$ (e.g., 1, 2, 0.5, -3, $\sqrt{2}$, etc.), there are infinitely many possible sets of direction ratios for any given line.

Therefore, Statement 1 is correct.

Statement 2: It is possible for certain lines that the sum of the squares of direction cosines can be equal to the sum of its direction cosines.

The statement says that for "certain lines", the following equality holds:

Sum of squares of direction cosines = Sum of direction cosines

We know the fundamental property of direction cosines $(l, m, n)$:

$\textrm{l}^2 + \textrm{m}^2 + \textrm{n}^2 = 1$

The statement implies that for certain lines, $1 = l + m + n$.

We need to check if there exist direction cosines $(l, m, n)$ that satisfy both conditions:

1. $\textrm{l}^2 + \textrm{m}^2 + \textrm{n}^2 = 1$
2. $\textrm{l} + \textrm{m} + \textrm{n} = 1$

Let's consider lines that are parallel to the coordinate axes.

• For a line parallel to the x-axis, the direction cosines are $(1, 0, 0)$.
  • Sum of squares: $1^2 + 0^2 + 0^2 = 1$.
  • Sum of direction cosines: $1 + 0 + 0 = 1$.
  • Here, $1 = 1$. The condition holds.
• For a line parallel to the y-axis, the direction cosines are $(0, 1, 0)$.
  • Sum of squares: $0^2 + 1^2 + 0^2 = 1$.
  • Sum of direction cosines: $0 + 1 + 0 = 1$.
  • Here, $1 = 1$. The condition holds.
• For a line parallel to the z-axis, the direction cosines are $(0, 0, 1)$.
  • Sum of squares: $0^2 + 0^2 + 1^2 = 1$.
  • Sum of direction cosines: $0 + 0 + 1 = 1$.
  • Here, $1 = 1$. The condition holds.

Since there exist lines (specifically, those parallel to the coordinate axes) for which the sum of the squares of direction cosines is equal to the sum of the direction cosines (both equal to 1), Statement 2 is correct.

Conclusion

Both Statement 1 and Statement 2 are correct.

Revision Table: Direction Ratios and Cosines

Additional Information: Lines in 3D Space

Understanding lines in three-dimensional space is fundamental in vector algebra and coordinate geometry. Lines are typically defined by a point they pass through and their direction.

• The direction of a line is often represented by its direction ratios or direction cosines.
• If $(x_1, y_1, z_1)$ is a point on the line and $(a, b, c)$ are its direction ratios, the equation of the line can be written in symmetric form as $\frac{\textrm{x} - \textrm{x}_1}{\textrm{a}} = \frac{\textrm{y} - \textrm{y}_1}{\textrm{b}} = \frac{\textrm{z} - \textrm{z}_1}{\textrm{c}}$.
• If the direction is given by direction cosines $(l, m, n)$, the symmetric form is $\frac{\textrm{x} - \textrm{x}_1}{l} = \frac{\textrm{y} - \textrm{y}_1}{m} = \frac{\textrm{z} - \textrm{z}_1}{n}$.
• Direction ratios are essentially the components of any vector parallel to the line. If $\vec{\textrm{v}} = a\hat{\textrm{i}} + b\hat{\textrm{j}} + c\hat{\textrm{k}}$ is a vector parallel to the line, then $(a, b, c)$ are direction ratios.
• Direction cosines are the components of a unit vector in the direction of the line. If $\hat{\textrm{v}} = l\hat{\textrm{i}} + m\hat{\textrm{j}} + n\hat{\textrm{k}}$ is a unit vector parallel to the line, then $(l, m, n)$ are the direction cosines, and $|\hat{\textrm{v}}|^2 = l^2 + m^2 + n^2 = 1$.