If `overset(rarr)A=2 hat i + 3 hat j- hat k` and `overset(rarr)B=-hat i+3 hat j +4 hat k` and then projection of `overset(rarr)A` on `overset(rarr)B` will be :

Options

A. `(3)/sqrt(13)`
B. `(3)/sqrt(26)`
C. `sqrt(3)/(26)`
D. `sqrt(3)/(13)`

Explanation

To find the projection of vector \(\overset{\rarr}{A}\) on vector \(\overset{\rarr}{B}\), we can use the formula for the projection of one vector onto another. The projection of vector \(\overset{\rarr}{A}\) onto vector \(\overset{\rarr}{B}\) is given by:

\[ \text{proj}_{\overset{\rarr}{B}} \overset{\rarr}{A} = \frac{\overset{\rarr}{A} \cdot \overset{\rarr}{B}}{|\overset{\rarr}{B}|^2} \overset{\rarr}{B} \]

### Step 1: Calculate the dot product \(\overset{\rarr}{A} \cdot \overset{\rarr}{B}\)

Given:
\[ \overset{\rarr}{A} = 2 \hat{i} + 3 \hat{j} - \hat{k} \]
\[ \overset{\rarr}{B} = -\hat{i} + 3 \hat{j} + 4 \hat{k} \]

The dot product is calculated as follows:
\[ \overset{\rarr}{A} \cdot \overset{\rarr}{B} = (2)(-1) + (3)(3) + (-1)(4) \]
\[ = -2 + 9 - 4 \]
\[ = 3 \]

### Step 2: Calculate the magnitude of vector \(\overset{\rarr}{B}\)

The magnitude of vector \(\overset{\rarr}{B}\) is given by:
\[ |\overset{\rarr}{B}| = \sqrt{(-1)^2 + (3)^2 + (4)^2} \]
\[ = \sqrt{1 + 9 + 16} \]
\[ = \sqrt{26} \]

### Step 3: Calculate \(|\overset{\rarr}{B}|^2\)

Now, we need \(|\overset{\rarr}{B}|^2\):
\[ |\overset{\rarr}{B}|^2 = 26 \]

### Step 4: Substitute into the projection formula

Now, substitute the values into the projection formula:
\[ \text{proj}_{\overset{\rarr}{B}} \overset{\rarr}{A} = \frac{3}{26} \overset{\rarr}{B} \]

### Step 5: Write the final projection vector

Substituting \(\overset{\rarr}{B}\):
\[ \text{proj}_{\overset{\rarr}{B}} \overset{\rarr}{A} = \frac{3}{26} (-\hat{i} + 3\hat{j} + 4\hat{k}) \]
\[ = -\frac{3}{26} \hat{i} + \frac{9}{26} \hat{j} + \frac{12}{26} \hat{k} \]

### Step 6: Magnitude of the projection

The magnitude of the projection is:
\[ \left|\text{proj}_{\overset{\rarr}{B}} \overset{\rarr}{A}\right| = \frac{3}{\sqrt{26}} \]

### Final Answer

Thus, the projection of vector \(\overset{\rarr}{A}\) on vector \(\overset{\rarr}{B}\) is:
\[ \frac{3}{\sqrt{26}} \]

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