What is the anlge between the line `(x+1)/(2) =(2y-1)/(4)=(2-z)/(4)` and the plane 2x+y-2z+4=0

Answer

To find the angle between the line given by the equations \((x+1)/2 = (2y-1)/4 = (2-z)/4\) and the plane defined by the equation \(2x + y - 2z + 4 = 0\), we can follow these steps:

### Step 1: Identify the Direction Ratios of the Line
The line is given in the symmetric form. We can rewrite it in terms of parameters. Let \(t\) be the parameter:

\[ x = 2t - 1, \quad y = \frac{t + 1}{2}, \quad z = 2 - 4t \]

From this, we can extract the direction ratios of the line, which are \(2, 4, -4\).

### Step 2: Identify the Normal Vector of the Plane
The equation of the plane is given as \(2x + y - 2z + 4 = 0\). The normal vector \(\mathbf{n}\) of the plane can be directly obtained from the coefficients of \(x\), \(y\), and \(z\):

\[ \mathbf{n} = (2, 1, -2) \]

### Step 3: Calculate the Angle Between the Line and the Normal Vector of the Plane
To find the angle \(\theta\) between the line and the plane, we first need to find the angle between the direction ratios of the line and the normal vector of the plane.

Using the formula for the cosine of the angle between two vectors:

\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \]

where \(\mathbf{a} = (2, 4, -4)\) (direction ratios of the line) and \(\mathbf{b} = (2, 1, -2)\) (normal vector of the plane).

### Step 4: Compute the Dot Product
The dot product \(\mathbf{a} \cdot \mathbf{b}\) is calculated as follows:

\[ \mathbf{a} \cdot \mathbf{b} = 2 \cdot 2 + 4 \cdot 1 + (-4) \cdot (-2) = 4 + 4 + 8 = 16 \]

### Step 5: Calculate the Magnitudes of the Vectors
Now, we calculate the magnitudes of both vectors:

\[ |\mathbf{a}| = \sqrt{2^2 + 4^2 + (-4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6 \]

\[ |\mathbf{b}| = \sqrt{2^2 + 1^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

### Step 6: Substitute into the Cosine Formula
Now we can substitute these values into the cosine formula:

\[ \cos \theta = \frac{16}{6 \cdot 3} = \frac{16}{18} = \frac{8}{9} \]

### Step 7: Find the Angle Between the Line and the Plane
The angle \(\phi\) between the line and the plane is given by:

\[ \phi = 90^\circ - \theta \]

To find \(\theta\), we can use \(\cos^{-1}\):

\[ \theta = \cos^{-1}\left(\frac{8}{9}\right) \]

Thus, the angle between the line and the plane is:

\[ \phi = 90^\circ - \cos^{-1}\left(\frac{8}{9}\right) \]

### Conclusion
The angle between the line and the plane is given by \(90^\circ - \theta\).