The position vectors of points A and B are `hati - hatj + 3hatk and 3hati + 3hatj - hatk` respectively. The equation of a plane is `vecr cdot (5hati + 2hatj - 7hatk)= 0` The points A and B

Options

A. Lie on the plane
B. Are on the same side of the plane
C. Are on the opposite sides of the plane
D. Nothing can be said

Answer

To determine the relationship of points A and B with respect to the given plane, we will follow these steps:

### Step 1: Identify the position vectors of points A and B
The position vectors are given as:
- Point A: \(\vec{r_A} = \hat{i} - \hat{j} + 3\hat{k}\)
- Point B: \(\vec{r_B} = 3\hat{i} + 3\hat{j} - \hat{k}\)

### Step 2: Write the equation of the plane
The equation of the plane is given by:
\[ \vec{r} \cdot (5\hat{i} + 2\hat{j} - 7\hat{k}) = 0 \]
Here, the normal vector \(\vec{n}\) to the plane is \(5\hat{i} + 2\hat{j} - 7\hat{k}\).

### Step 3: Substitute the position vector of point A into the plane equation
We substitute \(\vec{r_A}\) into the plane equation:
\[ \vec{r_A} \cdot (5\hat{i} + 2\hat{j} - 7\hat{k}) = (\hat{i} - \hat{j} + 3\hat{k}) \cdot (5\hat{i} + 2\hat{j} - 7\hat{k}) \]
Calculating the dot product:
\[ = 1 \cdot 5 + (-1) \cdot 2 + 3 \cdot (-7) = 5 - 2 - 21 = -18 \]
Since \(-18 < 0\), we have:
\[ \vec{r_A} \cdot \vec{n} < 0 \]

### Step 4: Substitute the position vector of point B into the plane equation
Now we substitute \(\vec{r_B}\) into the plane equation:
\[ \vec{r_B} \cdot (5\hat{i} + 2\hat{j} - 7\hat{k}) = (3\hat{i} + 3\hat{j} - \hat{k}) \cdot (5\hat{i} + 2\hat{j} - 7\hat{k}) \]
Calculating the dot product:
\[ = 3 \cdot 5 + 3 \cdot 2 + (-1) \cdot (-7) = 15 + 6 + 7 = 28 \]
Since \(28 > 0\), we have:
\[ \vec{r_B} \cdot \vec{n} > 0 \]

### Step 5: Analyze the results
From the calculations:
- \(\vec{r_A} \cdot \vec{n} < 0\)
- \(\vec{r_B} \cdot \vec{n} > 0\)

This indicates that point A lies on one side of the plane while point B lies on the opposite side.

### Conclusion
The points A and B are on opposite sides of the plane.