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. (a)`(3)/sqrt(13)`
B. (b)`(3)/sqrt(26)`
C. (c)`sqrt(3)/(26)`
D. (d)`sqrt(3)/(13)`

Explanation

To find the projection of vector **A** on vector **B**, we can follow these steps:

### Step 1: Identify the vectors
Given:
\[ \overset{\rarr}{A} = 2 \hat{i} + 3 \hat{j} - \hat{k} \]
\[ \overset{\rarr}{B} = -\hat{i} + 3 \hat{j} + 4 \hat{k} \]

### Step 2: Use the projection formula
The formula for the projection of vector **A** on vector **B** is given by:
\[ \text{Projection of } \overset{\rarr}{A} \text{ on } \overset{\rarr}{B} = \frac{\overset{\rarr}{A} \cdot \overset{\rarr}{B}}{|\overset{\rarr}{B}|^2} \overset{\rarr}{B} \]

### Step 3: Calculate the dot product \( \overset{\rarr}{A} \cdot \overset{\rarr}{B} \)
To find the dot product, we multiply the corresponding components of **A** and **B**:
\[ \overset{\rarr}{A} \cdot \overset{\rarr}{B} = (2)(-1) + (3)(3) + (-1)(4) \]
Calculating this gives:
\[ = -2 + 9 - 4 = 3 \]

### Step 4: Calculate the magnitude of vector **B**
The magnitude of vector **B** is calculated as follows:
\[ |\overset{\rarr}{B}| = \sqrt{(-1)^2 + (3)^2 + (4)^2} \]
Calculating this gives:
\[ = \sqrt{1 + 9 + 16} = \sqrt{26} \]

### Step 5: Substitute into the projection formula
Now we substitute the values we found into the projection formula:
\[ \text{Projection of } \overset{\rarr}{A} \text{ on } \overset{\rarr}{B} = \frac{3}{|\overset{\rarr}{B}|^2} \overset{\rarr}{B} \]
Calculating \( |\overset{\rarr}{B}|^2 \):
\[ |\overset{\rarr}{B}|^2 = 26 \]
Thus,
\[ \text{Projection of } \overset{\rarr}{A} \text{ on } \overset{\rarr}{B} = \frac{3}{26} \overset{\rarr}{B} \]

### Step 6: Final answer
The projection of vector **A** on vector **B** is:
\[ \frac{3}{26} (-\hat{i} + 3\hat{j} + 4\hat{k}) \]