The direction-cosines of te vector `vec(a) = hati - hatj - 2 hatk ` are ,

Options

A. `lt 1,-1, -2 gt`
B. `((1)/(sqrt(6)), - (1)/(sqrt(6)), (-2)/(sqrt(6)) )`
C. `( (1)/(4), -(1)/(4), (-2)/(4))`
D. `( sqrt((1)/(6)), - sqrt((1)/(6)), - sqrt((2)/(6)) )`.

Answer

To find the direction cosines of the vector \(\vec{a} = \hat{i} - \hat{j} - 2\hat{k}\), we follow these steps:

### Step 1: Identify the components of the vector
The vector \(\vec{a}\) can be expressed in terms of its components:
- \(A = 1\) (coefficient of \(\hat{i}\))
- \(B = -1\) (coefficient of \(\hat{j}\))
- \(C = -2\) (coefficient of \(\hat{k}\))

### Step 2: Calculate the magnitude of the vector
The magnitude \(|\vec{a}|\) of the vector is given by the formula:
\[ |\vec{a}| = \sqrt{A^2 + B^2 + C^2} \]
Substituting the values:
\[ |\vec{a}| = \sqrt{1^2 + (-1)^2 + (-2)^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \]

### Step 3: Calculate the direction cosines
The direction cosines \(l\), \(m\), and \(n\) are given by the formulas:
\[ l = \frac{A}{|\vec{a}|}, \quad m = \frac{B}{|\vec{a}|}, \quad n = \frac{C}{|\vec{a}|} \]
Substituting the values we found:
\[ l = \frac{1}{\sqrt{6}}, \quad m = \frac{-1}{\sqrt{6}}, \quad n = \frac{-2}{\sqrt{6}} \]

### Step 4: Write the final result
The direction cosines of the vector \(\vec{a}\) are:
\[ \left(\frac{1}{\sqrt{6}}, \frac{-1}{\sqrt{6}}, \frac{-2}{\sqrt{6}}\right) \]

### Summary
Thus, the direction cosines of the vector \(\vec{a} = \hat{i} - \hat{j} - 2\hat{k}\) are \(\left(\frac{1}{\sqrt{6}}, \frac{-1}{\sqrt{6}}, \frac{-2}{\sqrt{6}}\right)\).

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