If `a=hati+4hatj+2sqrt(2)hatk` and `b=(hati+hatj)sqrt(2)` then find component of a perpendicular to `b`

Options

A. `5/(sqrt(2))`
B. `9/(sqrt(2))`
C. `7/(sqrt(2))`
D. None of these

Answer

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

### Step 1: Identify the vectors
Given:
- **a** = \( \hat{i} + 4\hat{j} + 2\sqrt{2}\hat{k} \)
- **b** = \( (\hat{i} + \hat{j})\sqrt{2} \)

### Step 2: Calculate the dot product of **a** and **b**
The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = (1)(\sqrt{2}) + (4)(\sqrt{2}) + (2\sqrt{2})(0) \]
\[ = \sqrt{2} + 4\sqrt{2} + 0 = 5\sqrt{2} \]

### Step 3: Calculate the magnitude of vector **b**
The magnitude of vector **b** is calculated as:
\[ |\mathbf{b}| = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \]

### Step 4: Calculate the component of **a** along **b**
The component of **a** along **b** is given by:
\[ \text{Component of } \mathbf{a} \text{ along } \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]
Substituting the values we found:
\[ \text{Component of } \mathbf{a} \text{ along } \mathbf{b} = \frac{5\sqrt{2}}{2} \]

### Step 5: Calculate the magnitude of vector **a**
The magnitude of vector **a** is calculated as:
\[ |\mathbf{a}| = \sqrt{(1)^2 + (4)^2 + (2\sqrt{2})^2} \]
Calculating each term:
\[ = \sqrt{1 + 16 + 8} = \sqrt{25} = 5 \]

### Step 6: Calculate the component of **a** perpendicular to **b**
The component of **a** perpendicular to **b** can be calculated using the formula:
\[ \text{Component of } \mathbf{a} \text{ perpendicular to } \mathbf{b} = \sqrt{|\mathbf{a}|^2 - \left(\text{Component of } \mathbf{a} \text{ along } \mathbf{b}\right)^2} \]
Substituting the values:
\[ = \sqrt{5^2 - \left(\frac{5\sqrt{2}}{2}\right)^2} \]
Calculating further:
\[ = \sqrt{25 - \frac{25 \cdot 2}{4}} = \sqrt{25 - 12.5} = \sqrt{12.5} = \frac{5\sqrt{2}}{2} \]

### Final Answer
The component of vector **a** perpendicular to vector **b** is \( \frac{5\sqrt{2}}{2} \).

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