Finding Direction Cosines from a Point in Parametric Form

The question asks us to find the direction cosines of a line, given a point that lies on it. The coordinates of the point are given in terms of a parameter 'p'.

Understanding the Parametric Form of a Line

A line in 3D space can be represented parametrically. If a point on the line is given as \( (x(p), y(p), z(p)) \), where \(x(p), y(p), z(p)\) are linear functions of 'p', this represents a line. The general form of a parametric point on a line is \( (x_0 + ap, y_0 + bp, z_0 + cp) \), where \( (x_0, y_0, z_0) \) is a fixed point on the line and \( (a, b, c) \) are the direction ratios of the line.

In this problem, the point is given as \( (p + 1, p - 3, \sqrt{2}p) \). We can rewrite this as:

• \( x = 1 \cdot p + 1 \)
• \( y = 1 \cdot p - 3 \)
• \( z = \sqrt{2} \cdot p + 0 \)

Comparing this to the general form \( (x_0 + ap, y_0 + bp, z_0 + cp) \), we can identify the components:

• The fixed point \( (x_0, y_0, z_0) \) is \( (1, -3, 0) \) (when \(p=0\)).
• The coefficients of 'p' are the direction ratios \( (a, b, c) \).

Thus, the direction ratios of the line are \( (a, b, c) = (1, 1, \sqrt{2}) \).

Calculating Direction Cosines

Direction cosines \( (l, m, n) \) of a line are the cosines of the angles made by the line with the positive x, y, and z axes, respectively. They are related to the direction ratios \( (a, b, c) \) by the formula:

\( l = \frac{a}{\sqrt{a^2 + b^2 + c^2}} \)

\( m = \frac{b}{\sqrt{a^2 + b^2 + c^2}} \)

\( n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} \)

First, let's calculate the magnitude of the direction ratios vector \( \sqrt{a^2 + b^2 + c^2} \):

\( \sqrt{1^2 + 1^2 + (\sqrt{2})^2} = \sqrt{1 + 1 + 2} = \sqrt{4} = 2 \)

Now we can calculate the direction cosines:

• \( l = \frac{a}{\sqrt{a^2 + b^2 + c^2}} = \frac{1}{2} \)
• \( m = \frac{b}{\sqrt{a^2 + b^2 + c^2}} = \frac{1}{2} \)
• \( n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}} \)

So, the direction cosines of the line are \( \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right) \).

Comparing with Options

Let's compare our calculated direction cosines with the given options:

• Option 1: \( \left(\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\right) \)
• Option 2: \( \left(\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}\right) \)
• Option 3: \( \left(\frac{1}{\sqrt{2}},\frac{1}{2},-\frac{1}{2}\right) \)

Our calculated direction cosines \( \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right) \) match Option 1.

Therefore, the direction cosines of the line are \( \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right) \).

Revision Table: Direction Cosines and Ratios

Additional Information on Lines in 3D Space

A line in three-dimensional space can be described in various ways:

• Vector Form: \( \mathbf{r} = \mathbf{a} + t\mathbf{v} \), where \( \mathbf{a} \) is the position vector of a point on the line, \( \mathbf{v} \) is a direction vector parallel to the line, and \( t \) is a scalar parameter.
• Parametric Form: If \( \mathbf{a} = (x_0, y_0, z_0) \) and \( \mathbf{v} = (a, b, c) \), the vector form expands to \( (x, y, z) = (x_0, y_0, z_0) + t(a, b, c) \), which gives the parametric equations:
  • \( x = x_0 + at \)
  • \( y = y_0 + bt \)
  • \( z = z_0 + ct \)
  The coefficients \( (a, b, c) \) are the direction ratios. In our problem, the parameter is 'p' instead of 't'.
• Cartesian Form (Symmetric Form): By eliminating the parameter 't' from the parametric equations (assuming \( a, b, c \neq 0 \)), we get \( \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \).

Direction ratios \( (a, b, c) \) determine the direction of the line. Any set of numbers proportional to \( (a, b, c) \) also represents the direction ratios of the same line. Direction cosines \( (l, m, n) \) are unique for a given directed line (or two sets \( \pm(l, m, n) \) for an undirected line).