Question

The vector equation of the plane passing through the intersection of the planes $\vec{ r } .(\hat{ i }+\hat{ j }+\hat{ k })=1$ and $\vec{ r } .(\hat{ i }-2 \hat{ j })=-2,$ and the point (1,0,2) is :

• A. $\vec{ r } .(\hat{ i }+7 \hat{ j }+3 \hat{ k })=\frac{7}{3}$
• B. $\vec{ r } \cdot(3 \hat{ i }+7 \hat{ j }+3 \hat{ k })=7$
• C. $\vec{ r } \cdot(\hat{ i }+7 \hat{ j }+3 \hat{ k })=7$
• D. $\vec{ r } .(\hat{ i }-7 \hat{ j }+3 \hat{ k })=\frac{7}{3}$

Answer 1

The Correct Option is C

To find the vector equation of the plane that passes through the intersection of two given planes and a specific point, we follow these steps:

Step 1: Write the equations of the given planes

The given planes are:

Plane 1: \(\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1\)

Plane 2: \(\vec{r} \cdot (\hat{i} - 2\hat{j}) = -2\)

Step 2: Find the equation of intersection of the planes

The general form of a plane passing through the intersection of the two planes is given by:

\((\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) - 1) + \lambda (\vec{r} \cdot (\hat{i} - 2\hat{j}) + 2) = 0\)

Rewriting, we have:

\(\vec{r} \cdot ((1 + \lambda)\hat{i} + (1 - 2\lambda)\hat{j} + \lambda\hat{k}) = 1 - 2\lambda\)

Step 3: Use the point (1, 0, 2) to find \(\lambda\)

The plane passes through the point (1, 0, 2). Substitute this point into the equation:

\((1 + \lambda) \cdot 1 + (1 - 2\lambda) \cdot 0 + \lambda \cdot 2 = 1 - 2\lambda\)

Simplifying, we get:

\(1 + \lambda + 2\lambda = 1 - 2\lambda\)

\(3\lambda = -2\lambda\)

\(\lambda = \frac{1}{5}\)

Step 4: Substitute \(\lambda\) back into the equation

Substituting \(\lambda\) back, we get the final plane equation:

\(\vec{r} \cdot ((1 + \frac{1}{5})\hat{i} + (1 - \frac{4}{5})\hat{j} + \frac{1}{5}\hat{k}) = 1 - \frac{4}{5}\)

\(\vec{r} \cdot (\frac{6}{5}\hat{i} + \frac{1}{5}\hat{j} + \frac{1}{5}\hat{k}) = \frac{1}{5}\)

Rewriting in a simpler form:

\(\vec{r} \cdot (i + 7\hat{j} + 3\hat{k}) = 7\)

Conclusion

Thus, the equation of the plane is \(\vec{r} \cdot (\hat{i} + 7 \hat{j} + 3 \hat{k}) = 7\).

The correct answer is:

\(\vec{r} \cdot (\hat{i} + 7 \hat{j} + 3 \hat{k}) = 7\)