Question

The direction cosines of the vector \(\hat{i} + \hat{j} - 2\hat{k}\) are

• A. (1, 1, -2)
• B. \( \left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}} \right) \)
• C. \( \left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}} \right) \)
• D. \( \left( -\frac{1}{\sqrt{6}}, -\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}} \right) \)

Hint:

The direction ratios of a vector are simply its scalar components (a, b, c). To find the direction cosines, you just need to divide each direction ratio by the magnitude of the vector. Remember that the sum of the squares of the direction cosines is always 1: \(l^2 + m^2 + n^2 = 1\). This can be used for a quick verification.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Direction cosines of a vector are the cosines of the angles that the vector makes with the positive x, y, and z axes. They are also the components of the unit vector in the direction of the given vector.
Step 2: Key Formula or Approach:
For a vector \(\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}\): 1. Find the magnitude of the vector: \(|\vec{v}| = \sqrt{a^2 + b^2 + c^2}\).
2. The direction cosines (\(l, m, n\)) are given by: \[ l = \frac{a}{|\vec{v}|}, \quad m = \frac{b}{|\vec{v}|}, \quad n = \frac{c}{|\vec{v}|} \] Step 3: Detailed Explanation:
The given vector is \(\vec{v} = \hat{i} + \hat{j} - 2\hat{k}\).
The components of the vector are \(a = 1\), \(b = 1\), and \(c = -2\). These are the direction ratios.
1. Calculate the magnitude of \(\vec{v}\): \[ |\vec{v}| = \sqrt{1^2 + 1^2 + (-2)^2} \] \[ |\vec{v}| = \sqrt{1 + 1 + 4} = \sqrt{6} \] 2. Calculate the direction cosines: \[ l = \frac{a}{|\vec{v}|} = \frac{1}{\sqrt{6}} \] \[ m = \frac{b}{|\vec{v}|} = \frac{1}{\sqrt{6}} \] \[ n = \frac{c}{|\vec{v}|} = \frac{-2}{\sqrt{6}} \] So, the direction cosines are \( \left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}} \right) \).
Step 4: Final Answer:
The direction cosines of the vector \(\hat{i} + \hat{j} - 2\hat{k}\) are \( \left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}} \right) \).