Direction Cosines Explained

The problem asks us to determine the direction cosines of a line that passes through two specific points in three-dimensional space: \((-2, 3, -4)\) and \((1, 2, 3)\). Understanding direction cosines is crucial in 3D geometry as they define the orientation of a line with respect to the coordinate axes.

Direction Ratios Calculation

To find the direction cosines, we first need to determine the direction ratios of the line. If a line passes through two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), its direction ratios \((a, b, c)\) are given by the differences in their respective coordinates:

• Difference in x-coordinates: $a = x_2 - x_1$
• Difference in y-coordinates: $b = y_2 - y_1$
• Difference in z-coordinates: $c = z_2 - z_1$

Given the points \((-2, 3, -4)\) and \((1, 2, 3)\):

• $x_1 = -2$, $y_1 = 3$, $z_1 = -4$
• $x_2 = 1$, $y_2 = 2$, $z_2 = 3$

Now, let's calculate the direction ratios:

• $a = 1 - (-2) = 1 + 2 = 3$
• $b = 2 - 3 = -1$
• $c = 3 - (-4) = 3 + 4 = 7$

So, the direction ratios of the line are \((3, -1, 7)\).

Magnitude of Direction Ratios

Next, we need to find the magnitude (or length) of the vector represented by these direction ratios. If the direction ratios are \((a, b, c)\), the magnitude, denoted as $r$, is calculated using the formula:

$r = \sqrt{a^2 + b^2 + c^2}$

Using our calculated direction ratios \((3, -1, 7)\):

$r = \sqrt{3^2 + (-1)^2 + 7^2}$

$r = \sqrt{9 + 1 + 49}$

$r = \sqrt{59}$

Direction Cosines Determination

Finally, the direction cosines \((l, m, n)\) are found by dividing each direction ratio by the magnitude. The formulas are:

• $l = \frac{a}{r}$
• $m = \frac{b}{r}$
• $n = \frac{c}{r}$

Substituting the values we found:

• $l = \frac{3}{\sqrt{59}}$
• $m = \frac{-1}{\sqrt{59}}$
• $n = \frac{7}{\sqrt{59}}$

Therefore, the direction cosines of the line passing through the points \((-2, 3, -4)\) and \((1, 2, 3)\) are $\left(\frac{3}{\sqrt{59}}, \frac{-1}{\sqrt{59}}, \frac{7}{\sqrt{59}}\right)$.

This matches option 4 provided in the question.