Problem 21 - Precalculus with Limits

Question

Finding the Component Form of a Vector In Exercises \(13-24,\) find the component form and magnitude of the vector v. \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(1,3)} & {(-8,-9)}\end{array}\)

Step-by-Step Solution

Answer

The component form of the vector is \((-9, -12)\) and the magnitude of the vector is \(15\).

Step 1: Find the Component Form

Subtract the coordinates of the initial point from the terminal point to find the component form of the vector. Component form of a vector \(\mathbf{v}\) is given by \(\mathbf{v} = (x_2 - x_1, y_2 - y_1)\) where \(x_1, y_1\) are the coordinates of the initial point and \(x_2, y_2\) are the coordinates of the terminal point. Using this formula, the component form of the vector is \((-8 - 1, -9 - 3)\) which simplifies to \((-9, -12)\). The component form of the vector is \((-9, -12)\).

Step 2: Find the Magnitude

The magnitude of a vector \(\mathbf{v} = (a, b)\) is given by the formula \(\|\mathbf{v}\| = \sqrt{a^2 + b^2}\). Substituting \((-9, -12)\) into this formula, the magnitude of the vector is \(\sqrt{(-9)^2 + (-12)^2} = \sqrt{225}\). This equals \(15\). Therefore, the magnitude of the vector is \(15\).

Key Concepts

Vector Magnitude

When we talk about the magnitude of a vector, we're referring to the vector's length or size. This concept is foundational in the study of vectors in precalculus and has applications in various fields like physics, engineering, and computer science.

To find the magnitude of a vector, we use the Pythagorean theorem which is deeply rooted in coordinate geometry. Imagine the vector as the hypotenuse of a right triangle where the components of the vector represent the adjacent and opposite sides. The magnitude, denoted as \(\|\mathbf{v}\|\), can be calculated using the formula \[\sqrt{a^2 + b^2}\], where \(a\) and \(b\) are the vector's horizontal and vertical components, respectively.

For example, if we have a vector \(\mathbf{v}\) with components \(\mathbf{v} = (-9, -12)\), we compute the magnitude as \[\|\mathbf{v}\| = \sqrt{(-9)^2 + (-12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15\]. Thus, the vector's magnitude is 15. This numerical value represents the distance from the initial point to the terminal point of the vector in the plane, devoid of direction — essentially the 'length' of the vector in the coordinate space.

Vectors in Precalculus

In precalculus, vectors are a significant topic. They are objects that have both a magnitude (how long they are) and a direction (where they are pointing). Unlike simple numbers which only have magnitude (like distance), vectors provide more complex information that's crucial in fields that involve direction and force, such as physics.

One can represent vectors in several forms: in component form (usually for computations), as a graphical arrow (to visualize direction and length), or in i-j notation (for algebraic manipulation). The component form, which we use in the given exercise, is particularly useful when working with vectors in coordinate systems like the Cartesian plane. The steps we followed in finding the component form help streamline calculations related to vectors, such as adding and subtracting vectors, or even finding the distance between two points in space.

Understanding how to find the magnitude and direction of vectors in precalculus is not only a fundamental mathematical skill but also lays the groundwork for more advanced studies in science and engineering where vectors play a crucial role.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides the framework for plotting points, lines, and shapes on the Cartesian plane. It is essential for understanding and manipulating the position and size of geometric figures in terms of coordinates.

In the context of our exercise, we used principles of coordinate geometry to find the component form of a vector. To do this, we recognized that each point has an x (horizontal) and y (vertical) coordinate. The change in these coordinates signifies the direction and length of the vector. When we subtract the coordinates of the initial point from the terminal point, we're effectively measuring that change along each axis.

Coordinate geometry is integral when working with component forms because it allows us to visualize these concepts as physical movements along the axes. By borrowing techniques from coordinate geometry, such as plotting points and calculating distances using the Pythagorean theorem, we can better understand vector operations in precalculus.