If `abs(a)^(2)=8" and "a times (i+j+2k)=0` then the value of `a*(-i+j+4k)` is

Options

A. `4/sqrt(3)`
B. `16/sqrt(3)`
C. `8/sqrt(3)`
D. `1/sqrt(3)`

Answer

To solve the problem, we need to find the value of \( a \cdot (-i + j + 4k) \) given that \( |a|^2 = 8 \) and \( a \cdot (i + j + 2k) = 0 \).

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
- We have \( |a|^2 = 8 \), which implies that \( a \) is a vector with a magnitude of \( \sqrt{8} = 2\sqrt{2} \).
- The condition \( a \cdot (i + j + 2k) = 0 \) indicates that \( a \) is orthogonal to the vector \( (i + j + 2k) \).

2. **Expressing \( a \)**:
- Since \( a \) is orthogonal to \( (i + j + 2k) \), we can express \( a \) in terms of a scalar multiple of the vector \( (i + j + 2k) \).
- Let \( a = \lambda (i + j + 2k) \) for some scalar \( \lambda \).

3. **Finding the Magnitude of \( a \)**:
- The magnitude of \( a \) can be computed as follows:
\[ |a| = |\lambda| \cdot |(i + j + 2k)| \]
- The magnitude of \( (i + j + 2k) \) is:
\[ |(i + j + 2k)| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \]
- Therefore, we have:
\[ |a| = |\lambda| \cdot \sqrt{6} \]

4. **Setting Up the Equation**:
- From the condition \( |a|^2 = 8 \), we can write:
\[ |\lambda|^2 \cdot 6 = 8 \]
- This simplifies to:
\[ |\lambda|^2 = \frac{8}{6} = \frac{4}{3} \]
- Thus, we find:
\[ |\lambda| = \frac{2}{\sqrt{3}} \]

5. **Expressing \( a \) in Component Form**:
- Now substituting \( \lambda \) back into the expression for \( a \):
\[ a = \frac{2}{\sqrt{3}} (i + j + 2k) = \frac{2}{\sqrt{3}} i + \frac{2}{\sqrt{3}} j + \frac{4}{\sqrt{3}} k \]

6. **Calculating \( a \cdot (-i + j + 4k) \)**:
- Now we compute the dot product:
\[ a \cdot (-i + j + 4k) = \left( \frac{2}{\sqrt{3}} i + \frac{2}{\sqrt{3}} j + \frac{4}{\sqrt{3}} k \right) \cdot (-i + j + 4k) \]
- Expanding this:
\[ = \frac{2}{\sqrt{3}}(-1) + \frac{2}{\sqrt{3}}(1) + \frac{4}{\sqrt{3}}(4) \]
- This simplifies to:
\[ = -\frac{2}{\sqrt{3}} + \frac{2}{\sqrt{3}} + \frac{16}{\sqrt{3}} = \frac{16}{\sqrt{3}} \]

### Final Answer:
Thus, the value of \( a \cdot (-i + j + 4k) \) is \( \frac{16}{\sqrt{3}} \).