Find a vactor of magnitude of 5 units parallel to the resultant of vector `veca = 2 hati + 3hatj + hatk and vecb= (hati-2 hatj-hatk)`

Answer

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

### Step 1: Find the resultant vector \(\vec{R}\)
The resultant vector \(\vec{R}\) is given by the sum of the two vectors \(\vec{a}\) and \(\vec{b}\):
\[ \vec{R} = \vec{a} + \vec{b} = (2\hat{i} + 3\hat{j} + \hat{k}) + (\hat{i} - 2\hat{j} - \hat{k}) \]
Now, we combine the components:
\[ \vec{R} = (2 + 1)\hat{i} + (3 - 2)\hat{j} + (1 - 1)\hat{k} = 3\hat{i} + 1\hat{j} + 0\hat{k} \]
Thus, the resultant vector is:
\[ \vec{R} = 3\hat{i} + 1\hat{j} \]

### Step 2: Calculate the magnitude of the resultant vector \(\vec{R}\)
The magnitude of \(\vec{R}\) is calculated using the formula:
\[ |\vec{R}| = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \]

### Step 3: Find the unit vector in the direction of \(\vec{R}\)
The unit vector \(\hat{u}\) in the direction of \(\vec{R}\) is given by:
\[ \hat{u} = \frac{\vec{R}}{|\vec{R}|} = \frac{3\hat{i} + 1\hat{j}}{\sqrt{10}} = \frac{3}{\sqrt{10}}\hat{i} + \frac{1}{\sqrt{10}}\hat{j} \]

### Step 4: Scale the unit vector to have a magnitude of 5
To find a vector \(\vec{V}\) of magnitude 5 that is parallel to \(\vec{R}\), we multiply the unit vector \(\hat{u}\) by 5:
\[ \vec{V} = 5 \hat{u} = 5\left(\frac{3}{\sqrt{10}}\hat{i} + \frac{1}{\sqrt{10}}\hat{j}\right) = \frac{15}{\sqrt{10}}\hat{i} + \frac{5}{\sqrt{10}}\hat{j} \]

### Final Result
Thus, the vector of magnitude 5 units parallel to the resultant of \(\vec{a}\) and \(\vec{b}\) is:
\[ \vec{V} = \frac{15}{\sqrt{10}}\hat{i} + \frac{5}{\sqrt{10}}\hat{j} \]

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