Question

The direction cosines of the vector $\vec{a} = -2\hat{i} + \hat{j} - \hat{k}$ are

• A. \(\dfrac{2}{√6}, \dfrac{1}{√6},\dfrac{1}{√6}\)
• B. \(\dfrac{-2}{√6}, \dfrac{1}{√6},\dfrac{-1}{√6}\)
• C. \(\dfrac{-2}{√6}, \dfrac{-1}{√6},\dfrac{-1}{√6}\)
• D. \(\dfrac{-2}{√6}, \dfrac{1}{√6},\dfrac{1}{√6}\)
• E. \(\dfrac{-2}{√6}, \dfrac{-1}{√6},\dfrac{1}{√6}\)

Answer 1

The Correct Option is B

Step 1: Compute the magnitude of the vector \(\vec{a}\). Given \(\vec{a} = -2\hat{i} + \hat{j} - \hat{k}\), the magnitude \(|\vec{a}|\) is: \[ |\vec{a}| = \sqrt{(-2)^2 + (1)^2 + (-1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \]

Step 2: Find the direction cosines. The direction cosines \((l, m, n)\) are the cosines of the angles that the vector makes with the positive \(x\), \(y\), and \(z\) axes, respectively. They are given by: \[ l = \frac{a_x}{|\vec{a}|} = \frac{-2}{\sqrt{6}}, \quad m = \frac{a_y}{|\vec{a}|} = \frac{1}{\sqrt{6}}, \quad n = \frac{a_z}{|\vec{a}|} = \frac{-1}{\sqrt{6}} \]

Step 3: Match with the given options. The direction cosines are \(\left(\frac{-2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{-1}{\sqrt{6}}\right)\). 

Answer 2

Given 

\(a=-2i+j-k\)  is a vector and we need to find the direction of Cosine of this vector

So we can proceed as 

step1

\(|a|= √((-2)^{2}+1^{2}+(-1)^{2})\)

      \(=\)\(√6\)

step 2 

Direction of cosine of a vector are 

\(\dfrac{-2}{√6}, \dfrac{1}{√6},\dfrac{-1}{√6}\)