Y – 1 = -1/3 (x – 1) Applications Physics and Engineering Using the point-slope form of a line, y – y1 = m(x – x1), where m is the slope and (x1, y1) is the point, the equation of the normal line is:Ĭonsider the function f(x) = x², and we want to find the normal line at the point where x = 1.Īt x = 1, the slope of the tangent line is 2, so the slope of the normal line is -1/2.Ĭonsider the function f(x) = x³, and we want to find the normal line at the point where x = 1.Īt x = 1, the slope of the tangent line is 3, so the slope of the normal line is -1/3. Exercise Example 1Ĭonsider the function f(x) = 2x + 3, and we want to find the normal line at the point where x = 1.Īt x = 1, the slope of the tangent line is 2, so the slope of the normal line is the negative reciprocal, which is -1/2. This property of normal lines is central to geometric optics. Reflection of LightĪ light ray reflects off a surface so that the angle it makes with the normal line (the angle of incidence) equals the angle between the reflected ray and the normal line (the angle of reflection). The gradient vector is always normal to the level surfaces of the function. In the context of gradient vectors in multivariable calculus, the gradient of a scalar function at a point is a vector that points in the direction of the greatest rate of increase of the function at that point, and its magnitude is the rate of change in that direction. The curvature measures how fast the curve or surface changes direction.įor a curve in a plane, the curvature at a point is the reciprocal of the circle radius that best fits the curve near that point, and the center of this circle lies on the normal line. The normal line plays an integral role in calculating the curvature of a curve or surface. However, in many applications (like computer graphics), normal vectors are often normalized to have a length of 1 for the sake of simplicity and standardization. It can be any positive value and still be a normal vector. When dealing with normal vectors (normal lines in 3D), the normal vector’s magnitude (or length) is not standardized. Changing the point of contact will generally result in a different normal line. The normal line depends on the point of contact on the curve or surface. Read more Prime Polynomial: Detailed Explanation and Examples
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