Question

Find the direction cosines of a line joining two points \( (-2, 4, -5) \) and \( (1, 2, 3) \).

Hint:

Remember that direction cosines are the components of the unit vector in the direction of the line. A quick check is that the sum of the squares of the direction cosines should always be equal to 1: \( l^2 + m^2 + n^2 = 1 \).

Answer 1

Step 1: Understanding the Concept:
The direction cosines of a line are the cosines of the angles the line makes with the positive x, y, and z axes. To find them, we first need the direction ratios of the line, which can be found from the coordinates of the two given points. Then, we normalize these ratios by dividing by the magnitude of the direction vector.
Step 2: Key Formula or Approach:
Let the two points be \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \).
1. The direction ratios (a, b, c) are given by:
\( a = x_2 - x_1, b = y_2 - y_1, c = z_2 - z_1 \)
2. The magnitude of the vector PQ is \( r = \sqrt{a^2 + b^2 + c^2} \).
3. The direction cosines (l, m, n) are:
\( l = \frac{a}{r}, m = \frac{b}{r}, n = \frac{c}{r} \)
Step 3: Detailed Explanation or Calculation:
Let the given points be \( P(-2, 4, -5) \) and \( Q(1, 2, 3) \).
1. Find the direction ratios (a, b, c):
\[ a = 1 - (-2) = 3 \] \[ b = 2 - 4 = -2 \] \[ c = 3 - (-5) = 8 \] So, the direction ratios are \( \langle 3, -2, 8 \rangle \).
2. Find the magnitude (r):
\[ r = \sqrt{3^2 + (-2)^2 + 8^2} = \sqrt{9 + 4 + 64} = \sqrt{77} \] 3. Find the direction cosines (l, m, n):
\[ l = \frac{a}{r} = \frac{3}{\sqrt{77}} \] \[ m = \frac{b}{r} = \frac{-2}{\sqrt{77}} \] \[ n = \frac{c}{r} = \frac{8}{\sqrt{77}} \] Step 4: Final Answer:
The direction cosines of the line are \( \left( \frac{3}{\sqrt{77}}, \frac{-2}{\sqrt{77}}, \frac{8}{\sqrt{77}} \right) \).