Statement 1: Line `(x-1)/1=(y-0)/2=(z+2)/(-1)` lies in the plane `2x-3y-4z-10=0.` Statement 2: if line ` vec r= vec a+lambda vec b` lies in the plane ` vec rdot vec c=n(w h e r en` is scalar`),t h e n vec bdot vec c=0.`

Options

A. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
B. Both the Statements are true, but Statement 2 is not the correct explanation for Statement 1.
C. Statement 1 is true and Statement 2 is false.
D. Statement 1 is false and Statement 2 is true.

Explanation

To determine whether the statements are true, we will analyze each statement step by step.

### Step 1: Analyze Statement 1
The line is given by the equation:
\[ \frac{x-1}{1} = \frac{y-0}{2} = \frac{z+2}{-1} \]
This can be rewritten in parametric form:
- \( x = 1 + t \)
- \( y = 2t \)
- \( z = -2 - t \)

Where \( t \) is a parameter.

### Step 2: Identify the Direction Vector
From the parametric equations, we can identify the direction vector \( \vec{b} \) of the line:
\[ \vec{b} = \langle 1, 2, -1 \rangle \]

### Step 3: Identify a Point on the Line
We can also find a point on the line by setting \( t = 0 \):
- Point \( P(1, 0, -2) \)

### Step 4: Analyze the Plane Equation
The plane is given by the equation:
\[ 2x - 3y - 4z - 10 = 0 \]
The normal vector \( \vec{c} \) of the plane can be derived from the coefficients of \( x, y, z \):
\[ \vec{c} = \langle 2, -3, -4 \rangle \]

### Step 5: Check if the Line Lies in the Plane
For the line to lie in the plane, the point \( P(1, 0, -2) \) must satisfy the plane equation:
\[ 2(1) - 3(0) - 4(-2) - 10 = 2 + 0 + 8 - 10 = 0 \]
Since the equation holds true, the point lies in the plane.

### Step 6: Check the Dot Product Condition
To confirm that the line lies in the plane, we check the dot product of the direction vector \( \vec{b} \) and the normal vector \( \vec{c} \):
\[ \vec{b} \cdot \vec{c} = \langle 1, 2, -1 \rangle \cdot \langle 2, -3, -4 \rangle = (1)(2) + (2)(-3) + (-1)(-4) = 2 - 6 + 4 = 0 \]
Since the dot product is zero, the line is indeed perpendicular to the normal vector of the plane, confirming that the line lies in the plane.

### Conclusion for Statement 1
Statement 1 is **true**.

### Step 7: Analyze Statement 2
Statement 2 states that if a line \( \vec{r} = \vec{a} + \lambda \vec{b} \) lies in the plane represented by \( \vec{r} \cdot \vec{c} = n \), then \( \vec{b} \cdot \vec{c} = 0 \).

### Step 8: Explanation of Statement 2
In this context:
- \( \vec{a} \) is a point on the line.
- \( \vec{b} \) is the direction vector of the line.
- \( \vec{c} \) is the normal vector of the plane.

For the line to lie in the plane, the direction vector \( \vec{b} \) must be perpendicular to the normal vector \( \vec{c} \). This is confirmed by the dot product condition \( \vec{b} \cdot \vec{c} = 0 \).

### Conclusion for Statement 2
Statement 2 is also **true** and serves as a correct explanation for Statement 1.

### Final Answer
Both statements are correct, and Statement 2 correctly explains Statement 1.

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