The unit vectors parallel to the resultant vectors of `2hat(i)+4hat(j)-5hat(k) and hat(i)+2hat(j)+3hat(k)` is

Options

A. `(3hat(i)+6hat(j)-2hat(k))/(7)`
B. `(hat(i)+hat(j)+hat(k))/(sqrt(3))`
C. `(hat(i)+hat(j)+2hat(k))/(sqrt(6))`
D. `(-hat(i)-hat(j)+8hat(k))/(sqrt(69))`

Answer

To find the unit vector parallel to the resultant of the vectors \( \mathbf{a} = 2\hat{i} + 4\hat{j} - 5\hat{k} \) and \( \mathbf{b} = \hat{i} + 2\hat{j} + 3\hat{k} \), we can follow these steps:

### Step 1: Find the resultant vector
The resultant vector \( \mathbf{u} \) is obtained by adding the two vectors \( \mathbf{a} \) and \( \mathbf{b} \).

\[ \mathbf{u} = \mathbf{a} + \mathbf{b} = (2\hat{i} + 4\hat{j} - 5\hat{k}) + (\hat{i} + 2\hat{j} + 3\hat{k}) \]

### Step 2: Combine the components
Now, we combine the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) components:

\[ \mathbf{u} = (2 + 1)\hat{i} + (4 + 2)\hat{j} + (-5 + 3)\hat{k} \]
\[ \mathbf{u} = 3\hat{i} + 6\hat{j} - 2\hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector
Next, we need to find the magnitude of the vector \( \mathbf{u} \):

\[ |\mathbf{u}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} \]
\[ |\mathbf{u}| = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

### Step 4: Find the unit vector
The unit vector \( \hat{u} \) in the direction of \( \mathbf{u} \) is given by:

\[ \hat{u} = \frac{\mathbf{u}}{|\mathbf{u}|} = \frac{3\hat{i} + 6\hat{j} - 2\hat{k}}{7} \]

### Final Result
Thus, the unit vector parallel to the resultant vector is:

\[ \hat{u} = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \]

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