Problem 37 - Calculus for Biology and Medicine

Question

Find the equation of the line through \((1,-2)\) and perpendicular to \([4,1]^{\prime}\).

Step-by-Step Solution

Answer

The equation of the perpendicular line is \(y = -4x + 2\).

Step 1: Understand the Slope of a Vector

The vector \([4,1]'\) represents the direction of a line. The slope \(m_1\) of this line can be calculated from this vector as \(m_1 = \frac{y}{x} = \frac{1}{4}\). This is the slope of the line parallel to the given vector.

Step 2: Identify the Slope of the Perpendicular Line

The slope of a line perpendicular to another line with slope \(m_1\) is the negative reciprocal of \(m_1\). Therefore, the slope \(m_2\) of the line perpendicular to the given line \([4,1]'\) is \(m_2 = -\frac{1}{m_1} = -4\).

Step 3: Use Point-Slope Form to Find Equation

To find the equation of a line with slope \(-4\) passing through the point \((1,-2)\), use the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (1,-2)\) and slope \(m = -4\).

Step 4: Plugpoint into Point-Slope Formula

Substitute the values into the point-slope formula: \(y - (-2) = -4(x - 1)\). Simplify this to \(y + 2 = -4x + 4\).

Step 5: Rearrange into Slope-Intercept Form

To express the equation in slope-intercept form \(y = mx + b\), solve for \(y\):\[y = -4x + 4 - 2\] which simplifies to \[y = -4x + 2\].

Key Concepts

Slope of Vector

Vectors are like arrows. They have both a direction and a magnitude. When we talk about the slope of a vector in a 2D space, we are primarily concerned with its direction. The vector \([4, 1]\) tells us how far to move along the x-axis and y-axis.

To find the slope of a line that follows this direction, we use the formula for slope:

• Multiply the vector components: \(m = \frac{\text{change in } y}{\text{change in } x}\)
• In our example, the vector is \([4, 1]\). So break it down: the change in y is 1, and change in x is 4.
• Using the formula, the slope \(m = \frac{1}{4}\).

A vector's slope gives you an intuitive idea of how steep (or flat) the line heading this direction is.

Negative Reciprocal Slope

When two lines are perpendicular, their slopes are connected in a special way. They are negative reciprocals of each other.

What does that mean? If you have a slope \(m_1\), the perpendicular slope \(m_2\) can be found using these steps:

• First, find the reciprocal: if \(m_1 = \frac{1}{4}\), the reciprocal is \(\frac{4}{1}\) or simply 4.
• Next, make it negative: the negative reciprocal of 4 is \(-4\).
• This means our perpendicular slope \(m_2 = -4\).

Understanding negative reciprocal slopes is crucial. It helps in determining how lines intersect at exactly ninety degrees, like the corners of a square.

Point-Slope Form

The point-slope form is like a blueprint for building a line. It's a way to write the equation when you have one point the line passes through and the slope of the line.Here is the point-slope formula:

• Simplified, it looks like this: \(y - y_1 = m(x - x_1)\).
• In our scenario, you have a point \((1, -2)\) and a slope \(-4\).
• Plug these values into the formula to get \(y - (-2) = -4(x - 1)\).

Simplifying this will give you a clean line equation:

• Distribute \(-4\) over \((x - 1)\): \(y + 2 = -4x + 4\).
• Move constant terms to solve for \(y\): \(y = -4x + 2\).

This form is practical. It quickly translates known information into a usable format for graphing or further calculation.