The unit vector parallel to the resultant of the vectors A = 4i+3j + 6k and B = -i + 3j-8k is

Options

A. `1/7(3hati + 6hatj - 2hatk)`
B. `1/7(3hati + 6hatj + 2hatk)`
C. `1/49(3hati + 6hatj - 2hatk)`
D. `1/49(3hati - 6hatj + 2hatk)`

Answer

To find the unit vector parallel to the resultant of the vectors \( \mathbf{A} = 4\mathbf{i} + 3\mathbf{j} + 6\mathbf{k} \) and \( \mathbf{B} = -\mathbf{i} + 3\mathbf{j} - 8\mathbf{k} \), we can follow these steps:

### Step 1: Find the Resultant Vector
The resultant vector \( \mathbf{R} \) is given by the sum of the two vectors \( \mathbf{A} \) and \( \mathbf{B} \).

\[ \mathbf{R} = \mathbf{A} + \mathbf{B} \]

Substituting the values of \( \mathbf{A} \) and \( \mathbf{B} \):

\[ \mathbf{R} = (4\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}) + (-\mathbf{i} + 3\mathbf{j} - 8\mathbf{k}) \]

### Step 2: Combine Like Terms
Now, we combine the components of \( \mathbf{R} \):

\[ \mathbf{R} = (4 - 1)\mathbf{i} + (3 + 3)\mathbf{j} + (6 - 8)\mathbf{k} \]

Calculating each component:

\[ \mathbf{R} = 3\mathbf{i} + 6\mathbf{j} - 2\mathbf{k} \]

### Step 3: Calculate the Magnitude of the Resultant Vector
The magnitude \( |\mathbf{R}| \) of the resultant vector can be calculated using the formula:

\[ |\mathbf{R}| = \sqrt{(R_x)^2 + (R_y)^2 + (R_z)^2} \]

Substituting the components of \( \mathbf{R} \):

\[ |\mathbf{R}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} \]

Calculating the squares:

\[ |\mathbf{R}| = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

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

\[ \hat{\mathbf{R}} = \frac{\mathbf{R}}{|\mathbf{R}|} \]

Substituting \( \mathbf{R} \) and its magnitude:

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

This can be expressed as:

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

### Final Answer
Thus, the unit vector parallel to the resultant of the vectors \( \mathbf{A} \) and \( \mathbf{B} \) is:

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

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