Let a vector `veca =alpha hati + 2hatj + beta hatk` `(alpha, beta in R)`,`veca` lies in the plane of the vectors, ` vecb= hati + hatj` and `vecc= hati -hatj+4hatk`. If `veca` bisects the angle between `vecb and vecc`, then :

Options

A. (a)`vec a* veci +3 =0`
B. (b)`vec a*hati + 1=0`
C. (c)`vec a * hatk+2=0`
D. (d)`veca*hatk +4=0`

Answer

To solve the problem, we need to find the vector \(\vec{a}\) that bisects the angle between the vectors \(\vec{b}\) and \(\vec{c}\). Let's break down the steps to find \(\vec{a}\).

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

### Step 2: Calculate the magnitudes of \(\vec{b}\) and \(\vec{c}\)
1. Magnitude of \(\vec{b}\):
\[ |\vec{b}| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

2. Magnitude of \(\vec{c}\):
\[ |\vec{c}| = \sqrt{1^2 + (-1)^2 + 4^2} = \sqrt{1 + 1 + 16} = \sqrt{18} = 3\sqrt{2} \]

### Step 3: Use the angle bisector formula
The vector that bisects the angle between \(\vec{b}\) and \(\vec{c}\) can be expressed as:
\[ \vec{a} = k \left( \frac{\vec{b}}{|\vec{b}|} + \frac{\vec{c}}{|\vec{c}|} \right) \]
where \(k\) is a scalar.

### Step 4: Substitute the values
Substituting the values we calculated:
\[ \vec{a} = k \left( \frac{\hat{i} + \hat{j}}{\sqrt{2}} + \frac{\hat{i} - \hat{j} + 4\hat{k}}{3\sqrt{2}} \right) \]

### Step 5: Simplify the expression
Combine the two vectors:
\[ \vec{a} = k \left( \frac{1}{\sqrt{2}}(\hat{i} + \hat{j}) + \frac{1}{3\sqrt{2}}(\hat{i} - \hat{j} + 4\hat{k}) \right) \]

Finding a common denominator:
\[ \vec{a} = k \left( \frac{3}{3\sqrt{2}}(\hat{i} + \hat{j}) + \frac{1}{3\sqrt{2}}(\hat{i} - \hat{j} + 4\hat{k}) \right) \]
\[ = k \left( \frac{3\hat{i} + 3\hat{j} + \hat{i} - \hat{j} + 4\hat{k}}{3\sqrt{2}} \right) \]
\[ = k \left( \frac{(3+1)\hat{i} + (3-1)\hat{j} + 4\hat{k}}{3\sqrt{2}} \right) \]
\[ = k \left( \frac{4\hat{i} + 2\hat{j} + 4\hat{k}}{3\sqrt{2}} \right) \]

### Step 6: Set coefficients equal
From \(\vec{a} = \alpha \hat{i} + 2 \hat{j} + \beta \hat{k}\), we can equate coefficients:
- Coefficient of \(\hat{i}\): \(\alpha = \frac{4k}{3\sqrt{2}}\)
- Coefficient of \(\hat{j}\): \(2 = \frac{2k}{3\sqrt{2}}\)
- Coefficient of \(\hat{k}\): \(\beta = \frac{4k}{3\sqrt{2}}\)

### Step 7: Solve for \(k\)
From the equation \(2 = \frac{2k}{3\sqrt{2}}\):
\[ k = 2 \cdot \frac{3\sqrt{2}}{2} = 3\sqrt{2} \]

### Step 8: Substitute \(k\) back to find \(\alpha\) and \(\beta\)
Substituting \(k\) back:
- \(\alpha = \frac{4(3\sqrt{2})}{3\sqrt{2}} = 4\)
- \(\beta = \frac{4(3\sqrt{2})}{3\sqrt{2}} = 4\)

### Final Result
Thus, the vector \(\vec{a}\) becomes:
\[ \vec{a} = 4 \hat{i} + 2 \hat{j} + 4 \hat{k} \]