The vector equation of the line passing through the point `(-1,5,4)` and perpendicular to the plane `z=0` is

Options

A. `overset(to)( r) = - hat(i) +5 hat(j) + 4 hat(k) + lambda ( hat(i) + hat(j) )`
B. `overset(to)( r) = - hat(i) + 5 hat(j) + (4 + lambda ) hat(k)`
C. `overset(to)( r) = hat(i) - 5 hat(j) -4 hat(k) + lambda hat(k)`
D. `overset(to)( r) = lambda hat(k)`

Answer

To find the vector equation of the line passing through the point \((-1, 5, 4)\) and perpendicular to the plane \(z = 0\), we can follow these steps:

### Step 1: Identify the point and the direction vector
The line passes through the point \((-1, 5, 4)\). The direction vector of the line must be perpendicular to the plane \(z = 0\). The normal vector to the plane \(z = 0\) is given by \(\mathbf{n} = (0, 0, 1)\).

### Step 2: Write the general form of the vector equation of a line
The vector equation of a line can be expressed in the form:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
where \(\mathbf{a}\) is a position vector of a point on the line, \(\lambda\) is a scalar parameter, and \(\mathbf{b}\) is the direction vector of the line.

### Step 3: Substitute the point into the equation
Here, the position vector \(\mathbf{a}\) corresponding to the point \((-1, 5, 4)\) is:
\[ \mathbf{a} = -1 \mathbf{i} + 5 \mathbf{j} + 4 \mathbf{k} \]

### Step 4: Identify the direction vector
Since the line is perpendicular to the plane \(z = 0\), the direction vector \(\mathbf{b}\) is the normal vector of the plane, which is:
\[ \mathbf{b} = 0 \mathbf{i} + 0 \mathbf{j} + 1 \mathbf{k} = \mathbf{k} \]

### Step 5: Write the complete vector equation
Now substituting \(\mathbf{a}\) and \(\mathbf{b}\) into the general form:
\[ \mathbf{r} = (-1 \mathbf{i} + 5 \mathbf{j} + 4 \mathbf{k}) + \lambda (0 \mathbf{i} + 0 \mathbf{j} + 1 \mathbf{k}) \]

### Step 6: Simplify the equation
This simplifies to:
\[ \mathbf{r} = -1 \mathbf{i} + 5 \mathbf{j} + (4 + \lambda) \mathbf{k} \]

### Final Answer
Thus, the vector equation of the line is:
\[ \mathbf{r} = -1 \mathbf{i} + 5 \mathbf{j} + (4 + \lambda) \mathbf{k} \]

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