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IB DP Maths AA SL Study Notes

5.2.2 Normal Line

Definition

At any given point on a curve, the normal line is the line that stands perpendicular to the tangent line. While the tangent line offers a glimpse into the direction the curve is heading at a particular point, the normal line provides a contrasting view. It points directly away from the curve, forming a perfect right angle with the tangent.

To understand more about the tangent line, refer to the Equation of a Tangent Line page.

Equation of the Normal Line

To derive the equation of the normal line, one must follow a systematic approach:

1. Determine the Slope of the Tangent Line: The initial step involves finding the derivative of the function. This derivative provides the slope of the tangent line for any point on the curve.

2. Calculate the Slope of the Normal Line: After determining the slope of the tangent line, represented as mt, the slope of the normal line, mn, can be deduced using the relationship: mn = -1 divided by mt This relationship stems from the geometric property that the product of the slopes of two perpendicular lines equals -1.

3. Employ the Point-Slope Form: With the slope of the normal line in hand, you can then use the point-slope form to determine its equation: y - y1 = mn (x - x1) Here, (x1, y1) denotes the point of tangency.

For related concepts, you may also want to explore the Normal Distribution.

Detailed Example

To solidify our understanding, let's walk through a comprehensive example:

Function: Consider the curve defined by the function y = x cubed.

Objective: Our goal is to determine the equation of the normal line at the point (2,8).

Step 1: Derivative of the Function The derivative of y = x cubed is y' = 3x squared. This derivative indicates the slope of the tangent line for any given x-value. For x = 2, y' equals 12. Thus, the slope of the tangent line, mt, at the point (2,8) is 12.

Step 2: Slope of the Normal Line Using our established relationship between the slopes of the tangent and normal lines, we deduce: mn = -1 divided by 12

Step 3: Equation of the Normal Line Utilising the point-slope form with the point (2,8) and the slope mn = -1 divided by 12, we derive: y - 8 = -1 divided by 12 (x - 2) On simplifying, we get the equation: y = -1 divided by 12x + 25 divided by 3

For an example involving arc length, check out Arc Length.

Applications and Significance

The normal line isn't just a theoretical concept; it finds practical applications across various fields:

  • Physics: In optics, the normal line plays a pivotal role. When light rays strike a surface, the angle they form with the normal line, known as the angle of incidence, dictates their subsequent behaviour, be it reflection or refraction.
  • Engineering: For engineers, understanding forces acting on surfaces is paramount, especially during structural design. The normal line assists in determining the perpendicular components of forces on surfaces, crucial for stability and safety.
  • Geometry and Design: Geometric constructions often rely on the normal line, especially when crafting perpendicular bisectors or ensuring orthogonality in designs. To see how this applies to correlation, review the Correlation Coefficient.
  • Advanced Calculus: As one delves deeper into calculus, the normal line becomes integral to advanced topics like curvature, osculating circles, and more. Understanding exponential growth can also be crucial, as detailed in Exponential Equations.

FAQ

The slope of the normal line is the negative reciprocal of the slope of the tangent line. This relationship arises from the fact that the normal and tangent lines are perpendicular to each other. In coordinate geometry, two lines are perpendicular if the product of their slopes is -1. Therefore, if the slope of the tangent line is m, the slope of the normal line would be -1/m.

Yes, the normal line can be either horizontal or vertical, depending on the slope of the tangent line. If the tangent line is horizontal (slope = 0), the normal line will be vertical. Conversely, if the tangent line is vertical, the normal line will be horizontal. This is because the negative reciprocal of 0 is undefined, leading to a vertical normal line, and the negative reciprocal of an undefined slope (vertical tangent) is 0, resulting in a horizontal normal line.

If the tangent line is vertical, its slope is undefined. In such a case, the normal line will be horizontal. A horizontal line has a slope of 0 and can be represented by the equation y = k, where k is the y-coordinate of the point of tangency. So, if the point of tangency is (a,b), the equation of the normal line will simply be y = b.

Every curve has a tangent line at points where it is differentiable. Since the normal line is defined based on the tangent line, if a curve has a tangent line at a point, it will also have a normal line at that point. However, at points of non-differentiability (like cusps or sharp turns), the curve might not have a well-defined tangent. In such cases, the curve won't have a normal line at that specific point.

The normal line is perpendicular to the tangent line at any given point on a curve. This perpendicular relationship is crucial in various mathematical and physical contexts. For instance, in physics, when considering reflection of light on a surface, the angle of incidence and reflection are measured with respect to the normal. In geometry, the normal line can be used to construct perpendicular bisectors. Moreover, in calculus, the normal line can be instrumental in understanding the behaviour of a function, especially in the context of curvature and surface analysis.

Practice Questions

Determine the equation of the normal line to the curve y = x squared at the point (1,1).

To find the equation of the normal line to the curve y = x squared at the point (1,1), we first identify the slope of the tangent line at that point. The derivative of y = x squared is y' = 2x. Evaluating at x = 1, we get y' = 2. This value represents the slope of the tangent line. The slope of the normal line is the negative reciprocal of the slope of the tangent. Thus, the slope of the normal line, mn, is -1/2. Using the point-slope formula with the point (1,1) and the slope mn = -1/2, we derive the equation of the normal line as y = -1/2x + 3/2.

Find the equation of the normal line to the curve y = 3x cubed - 2x + 1 at the point (-1,6).

To determine the equation of the normal line to the curve y = 3x cubed - 2x + 1 at the point (-1,6), we first ascertain the slope of the tangent line at that point. The derivative of y = 3x cubed - 2x + 1 is y' = 9x squared - 2. At x = -1, y' equals 7. This value gives us the slope of the tangent line. The slope of the normal line is the negative reciprocal of the slope of the tangent. Therefore, the slope of the normal line, mn, is -1/7. Using the point-slope formula with the point (-1,6) and the slope mn = -1/7, we arrive at the equation of the normal line as y = -1/7x + 41/7.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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