Question

If \(\bar{a}=2\hat{i}+3\hat{j}-4\hat{k}\) and \(\bar{b}=\hat{i}+3\hat{j}+2\hat{k}\), then a unit vector in the direction of \(\bar a +\bar b\) is

• A. \(\frac{1}{6}(3\hat{i}+6\hat{j}-2\hat{k})\)
• B. \(\frac{1}{\sqrt{70}}(3\hat{i}+6\hat{j}-5\hat{k})\)
• C. \(\frac{1}{7}(3\hat{i}+6\hat{j}-2\hat{k})\)
• D. \(\frac{1}{\sqrt{50}}(3\hat{i}+6\hat{j}-3\hat{k})\)
• E. \(\frac{1}{\sqrt{6}}(\hat{i}+2\hat{j}-\hat{k})\)

Answer 1

The Correct Option is C

We are given two vectors: \[ \mathbf{a} = 2\hat{i} + 3\hat{j} - 4\hat{k}, \quad \mathbf{b} = \hat{i} + 3\hat{j} + 2\hat{k}. \] We need to find a unit vector in the direction of \( \mathbf{a} + \mathbf{b} \).
Step 1: Find \( \mathbf{a} + \mathbf{b} \) \[ \mathbf{a} + \mathbf{b} = (2\hat{i} + 3\hat{j} - 4\hat{k}) + (\hat{i} + 3\hat{j} + 2\hat{k}) \] Simplify: \[ \mathbf{a} + \mathbf{b} = (2 + 1)\hat{i} + (3 + 3)\hat{j} + (-4 + 2)\hat{k} \] \[ \mathbf{a} + \mathbf{b} = 3\hat{i} + 6\hat{j} - 2\hat{k}. \]
Step 2: Find the magnitude of \( \mathbf{a} + \mathbf{b} \) The magnitude of \( \mathbf{a} + \mathbf{b} \) is given by: \[ |\mathbf{a} + \mathbf{b}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} \] \[ |\mathbf{a} + \mathbf{b}| = \sqrt{9 + 36 + 4} = \sqrt{49} = 7. \]
Step 3: Find the unit vector The unit vector in the direction of \( \mathbf{a} + \mathbf{b} \) is given by: \[ \hat{u} = \frac{\mathbf{a} + \mathbf{b}}{|\mathbf{a} + \mathbf{b}|}. \] Substitute the values: \[ \hat{u} = \frac{1}{7} (3\hat{i} + 6\hat{j} - 2\hat{k}). \]

The correct option is (C) : \(\frac{1}{7}(3\hat{i}+6\hat{j}-2\hat{k})\)

Answer 2

We are given \(\bar{a} = 2\hat{i} + 3\hat{j} - 4\hat{k}\) and \(\bar{b} = \hat{i} + 3\hat{j} + 2\hat{k}\).

First, we find \(\bar{a} + \bar{b}\):

\(\bar{a} + \bar{b} = (2\hat{i} + 3\hat{j} - 4\hat{k}) + (\hat{i} + 3\hat{j} + 2\hat{k}) = (2+1)\hat{i} + (3+3)\hat{j} + (-4+2)\hat{k} = 3\hat{i} + 6\hat{j} - 2\hat{k}\)

Now, we find the magnitude of \(\bar{a} + \bar{b}\):

\(|\bar{a} + \bar{b}| = \sqrt{3^2 + 6^2 + (-2)^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7\)

To find a unit vector in the direction of \(\bar{a} + \bar{b}\), we divide the vector by its magnitude:

\(\frac{\bar{a} + \bar{b}}{|\bar{a} + \bar{b}|} = \frac{3\hat{i} + 6\hat{j} - 2\hat{k}}{7} = \frac{1}{7}(3\hat{i} + 6\hat{j} - 2\hat{k})\)

Therefore, a unit vector in the direction of \(\bar{a} + \bar{b}\) is \(\frac{1}{7}(3\hat{i} + 6\hat{j} - 2\hat{k})\).