Problem 24 - Precalculus with Limits

Question

In Exercises 13-24, find the component form and the magnitude of the vector \(\mathbf{v}\).'' Initial Point - \((-3, 11)\) Terminal Point - \((9, 40)\)

Step-by-Step Solution

Answer

The component form of the vector is \( \mathbf{v} = (12, 29)\) and the magnitude of the vector is \( \sqrt{985}\).

Step 1: Find the Component Form

The component form of the vector can be found by subtracting the coordinates of the initial point from the coordinates of the terminal point. This can be done using the formula \((x_2 - x_1, y_2 - y_1)\), where \((x_1, y_1)\) represents the initial point and \((x_2, y_2)\) represents the terminal point. Substituting the given points - initial point \((-3, 11)\) and terminal point \((9, 40)\) into the formula gives: \((9 - (-3), 40 - 11) = (9 + 3, 40 - 11) = (12, 29)\). Therefore, the component form of the vector is \( \mathbf{v} = (12, 29)\).

Step 2: Find the Magnitude (or Length) of the Vector

The magnitude of the vector can be calculated using the Pythagorean theorem. The formula for finding the magnitude of a vector \( \mathbf{v} = (a, b)\) is \(\sqrt{a^2 + b^2}\). Substituting \(a = 12\) and \(b = 29\) into the formula gives: \(\sqrt{12^2 + 29^2} = \sqrt{144 + 841} = \sqrt{985}\). The exact magnitude of the vector is \( \sqrt{985}\). If we need the decimal approximation, it would be close to 31.38, but usually in mathematics, keeping the exact square root form is preferred unless asked otherwise.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics, intimately related to the geometry of right-angled triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This theorem can be written as an equation: \(c^2 = a^2 + b^2\), where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the triangle's other two sides. This relation has a broad range of applications, including its use in calculating the magnitude of vectors in coordinate geometry.

Vector Components

Vector components are projections of a vector onto the axes of a coordinate system. In two dimensions, any vector can be represented by its components along the horizontal (x) and vertical (y) axes. For a vector with its tail at the origin and its head at point \( (x, y) \), its component form is simply \( (x, y) \).

When the vector's starting point, or 'tail', is not at the origin, as is often the case in problems involving displacement, the component form can be found by subtracting the coordinates of the initial point from the terminal point, as represented by \(\vec{v} = (x_2 - x_1, y_2 - y_1)\). These vector components are essential for various vector operations, including vector addition, subtraction, and finding the magnitude.

Magnitude of a Vector

The magnitude of a vector, often referred to as its length or norm, is a measure of how long the vector is. For a two-dimensional vector with components \(a\) and \(b\), the magnitude is found using the Pythagorean theorem applied to a right-angled triangle with sides of lengths \(a\) and \(b\), and the vector as the hypotenuse. The formula for magnitude is \(\sqrt{a^2 + b^2}\).

Understanding the magnitude is crucial for many physics problems where vectors represent quantities such as displacement, velocity, or force. The magnitude provides a scalar quantity representing the size of the vector independent of its direction.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This approach combines algebra and geometry to describe the position, the line segments, and shapes using points defined by their coordinates on the plane.

One common application is determining the position of points, dividing segments into parts, or calculating areas and volumes of geometric figures. Coordinate geometry simplifies many geometry problems by allowing them to be solved with algebraic methods, such as finding the distance between two points or the area of a triangle given its vertices' coordinates.