A point `(alpha, beta, gamma)` satisfies the equation of the plane `3x+4y+7z=3.` The value of `beta`, such that `vecp=alphahati+betahatj+gammahatk` satisfies the relation `hatjxx(hatjxxvecp)=vec0`, is equal to

Answer

To solve the problem step by step, we need to find the value of \( \beta \) for the point \( (\alpha, \beta, \gamma) \) that satisfies the equation of the plane \( 3x + 4y + 7z = 3 \) and the relation \( \hat{j} \times (\hat{j} \times \vec{p}) = \vec{0} \).

### Step 1: Substitute the point into the plane equation
The point \( (\alpha, \beta, \gamma) \) satisfies the equation of the plane:
\[ 3\alpha + 4\beta + 7\gamma = 3 \]

### Step 2: Understand the vector \( \vec{p} \)
The vector \( \vec{p} \) can be expressed as:
\[ \vec{p} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \]

### Step 3: Apply the vector triple product identity
We know that:
\[ \hat{j} \times (\hat{j} \times \vec{p}) = (\hat{j} \cdot \vec{p}) \hat{j} - (\hat{j} \cdot \hat{j}) \vec{p} \]
Since \( \hat{j} \cdot \hat{j} = 1 \), this simplifies to:
\[ \hat{j} \times (\hat{j} \times \vec{p}) = (\hat{j} \cdot \vec{p}) \hat{j} - \vec{p} \]

### Step 4: Calculate \( \hat{j} \cdot \vec{p} \)
Calculating the dot product:
\[ \hat{j} \cdot \vec{p} = \hat{j} \cdot (\alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}) = 0 \cdot \alpha + 1 \cdot \beta + 0 \cdot \gamma = \beta \]

### Step 5: Substitute into the equation
Substituting back, we have:
\[ \hat{j} \times (\hat{j} \times \vec{p}) = \beta \hat{j} - \vec{p} \]
Setting this equal to \( \vec{0} \):
\[ \beta \hat{j} - (\alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}) = \vec{0} \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ \beta \hat{j} - \beta \hat{j} - \alpha \hat{i} - \gamma \hat{k} = \vec{0} \]
This simplifies to:
\[ -\alpha \hat{i} - \gamma \hat{k} = \vec{0} \]
From this, we conclude:
\[ \alpha = 0 \quad \text{and} \quad \gamma = 0 \]

### Step 7: Substitute back into the plane equation
Substituting \( \alpha = 0 \) and \( \gamma = 0 \) into the plane equation:
\[ 3(0) + 4\beta + 7(0) = 3 \implies 4\beta = 3 \]

### Step 8: Solve for \( \beta \)
Solving for \( \beta \):
\[ \beta = \frac{3}{4} \]

Thus, the value of \( \beta \) is \( \frac{3}{4} \).