If `a = 4i + 6j and b = 3j+4k`, then the vector form of component of a along b is

Options

A. `(18)/(10sqrt((3)))(3j+4k)`
B. `(18)/(25) (3j+4k)`
C. `(18)/(sqrt((3)))(3j+4k)`
D. `3j+4k`

Answer

To find the vector form of the component of vector **a** along vector **b**, we can follow these steps:

### Step 1: Define the vectors
Given:
- **a** = 4i + 6j
- **b** = 3j + 4k

### Step 2: Calculate the dot product of **a** and **b**
The dot product of two vectors **a** and **b** is given by:
\[ \mathbf{a} \cdot \mathbf{b} = (4i + 6j) \cdot (3j + 4k) \]
We can calculate this as follows:
- The i component of **a** (4i) has no corresponding component in **b**, so it contributes 0 to the dot product.
- The j component of **a** (6j) and the j component of **b** (3j) contribute:
\[ 6 \cdot 3 = 18 \]
- The k component of **b** (4k) has no corresponding component in **a**, so it contributes 0 to the dot product.

Thus, the dot product is:
\[ \mathbf{a} \cdot \mathbf{b} = 0 + 18 + 0 = 18 \]

### Step 3: Calculate the magnitude of vector **b**
The magnitude of vector **b** is given by:
\[ |\mathbf{b}| = \sqrt{(0)^2 + (3)^2 + (4)^2} = \sqrt{0 + 9 + 16} = \sqrt{25} = 5 \]

### Step 4: Calculate the unit vector of **b**
The unit vector **b̂** in the direction of **b** is given by:
\[ \mathbf{\hat{b}} = \frac{\mathbf{b}}{|\mathbf{b}|} = \frac{(3j + 4k)}{5} = \frac{3}{5}j + \frac{4}{5}k \]

### Step 5: Calculate the component of **a** along **b**
The component of **a** along **b** is given by the formula:
\[ \text{Component of } \mathbf{a} \text{ along } \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \mathbf{\hat{b}} \]
Substituting the values we found:
\[ \text{Component of } \mathbf{a} \text{ along } \mathbf{b} = \frac{18}{5} \left( \frac{3}{5}j + \frac{4}{5}k \right) \]

### Step 6: Simplify the expression
\[ = \frac{18}{5} \cdot \frac{3}{5}j + \frac{18}{5} \cdot \frac{4}{5}k = \frac{54}{25}j + \frac{72}{25}k \]

### Final Result
The vector form of the component of **a** along **b** is:
\[ \frac{54}{25}j + \frac{72}{25}k \]