Problem 21 - Precalculus with Limits

Question

In Exercises 13-24, find the component form and the magnitude of the vector \(\mathbf{v}\).'' Initial Point - \((1, 3)\) Terminal Point - \((-8, -9)\)

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 of the Vector

In order to find the component form of the vector, subtract the coordinates of the initial point from the terminal point. The initial point is (1, 3) and the terminal point is (-8, -9). Therefore, the component form of the vector is (-8-1, -9-3), which simplifies to (-9, -12).

Step 2: Compute the Magnitude of the Vector

The magnitude of a vector can be calculated using the formula: \(\sqrt{{x^2 + y^2}}\) where x and y are the coordinates of the component form. Given the components (-9, -12) from step 1, the magnitude can be computed as follows: \(\sqrt{(-9)^2 + (-12)^2} = \sqrt{81 + 144}= \sqrt{225}\).

Step 3: Simplify the Magnitude of the Vector

Finally, the magnitude of the vector \(\mathbf{v}\) is the square root of 225, which is 15.

Key Concepts

Vector Component Form

Vectors have a special way of expressing their size and direction, known as the component form. To find this form, we must first establish an initial and a terminal point. These points, each with their own x- and y-coordinates, outline the path of the vector. For instance, if you have an initial point at \((1, 3)\) and a terminal point at \((-8, -9)\), you can derive the vector's journey by subtracting the coordinates of the initial point from those of the terminal point.

This process is detailed in the following manner:

• Subtract the x-coordinate of the initial point from the x-coordinate of the terminal point: \(-8 - 1 = -9\).
• Subtract the y-coordinate of the initial point from the y-coordinate of the terminal point: \(-9 - 3 = -12\).

Thus, the vector in component form becomes \((-9, -12)\). This expression allows us to understand the vector's position change from start to finish, showcasing both horizontal and vertical movements.

Vector Magnitude

The magnitude of a vector measures its length, giving a clear picture of its size without regard to direction. Imagine the magnitude as the distance from the vector’s initial point to its terminal point. To calculate this, the Pythagorean theorem is used. For a vector expressed in component form \((x, y)\), the formula for magnitude is:

\[ \sqrt{x^2 + y^2} \]

For example, consider the vector we discussed earlier with components \((-9, -12)\):

• Square the x-component: \((-9)^2 = 81\).
• Square the y-component: \((-12)^2 = 144\).
• Add the results: \(81 + 144 = 225\).
• Finally, take the square root: \(\sqrt{225} = 15\).

The result, \(15\), indicates the vector’s magnitude, offering insight into its "length" on the coordinate plane.

Coordinate Geometry

Coordinate geometry acts as a bridge between algebra and geometry by using coordinates to explore geometric properties. Vectors, points, lines, and shapes can all be described through their positions on a plane, identified by coordinate points. Consider a vector with an initial point \((1, 3)\) and a terminal point \((-8, -9)\). In coordinate geometry, these coordinates play a crucial role in illustrating the vector’s path.

Coordinate geometry provides tools to calculate distances, derive slopes, and determine geometric properties through simple operations like addition and subtraction. Here's how it connects to vectors:

• Initial and terminal points define a vector's direction and length.
• By subtracting coordinates, we access the vector’s component form, which reveals movement between points.

Using coordinate geometry, we successfully interpret and solve vector-related problems, translating geometric movements into algebraic expressions, and helping understand the complex interactions within the coordinate plane.