The angle between the line x−21=y−32=z−4−1  and the plane x + y + z = 4 is

The angle between a line and a plane can be found using the direction ratios of the line and the normal to the plane.

Step 1: Find the direction ratios of the line The given line is x−2 = y−3 = z−4 = -1. We can rewrite this as x/1 = y/1 = z/1 = -1. So, the direction ratios of the line are (1, 1, 1).

Step 2: Find the normal to the plane The given plane is x + y + z = 4. The coefficients of x, y, and z in the equation of the plane give the direction ratios of the normal to the plane. So, the direction ratios of the normal to the plane are (1, 1, 1).

Step 3: Find the angle between the line and the plane The angle θ between the line and the plane is given by the formula cos θ = (a1a2 + b1b2 + c1c2) / √[(a1² + b1² + c1²) * (a2² + b2² + c2²)], where (a1, b1, c1) are the direction ratios of the line and (a2, b2, c2) are the direction ratios of the normal to the plane.

Substituting the given values, we get cos θ = (11 + 11 + 1*1) / √[(1² + 1² + 1²) * (1² + 1² + 1²)] = 3 / √[3 * 3] = 3 / √9 = 3/3 = 1.

Therefore, the angle θ = cos⁻¹(1) = 0 degrees.

So, the angle between the given line and plane is 0 degrees. This means the line is parallel to the plane.