Problem 25 - Algebra and Trigonometry

Question

\(23-26\) Sketch representations of the given vector with initial points at \((0,0),(2,3),\) and \((-3,5)\). $$ \mathbf{u}=\langle- 7,2\rangle $$

Step-by-Step Solution

Answer

Draw the vector \(\langle -7,2\rangle\) from \((0,0)\) to \((-7,2)\), from \((2,3)\) to \((-5,5)\), and from \((-3,5)\) to \((-10,7)\).

Step 1: Understand the Problem

We need to sketch the vector \( \mathbf{u} = \langle -7, 2 \rangle \) starting from three different initial points: \((0,0), (2,3)\), and \((-3,5)\). This means we'll draw the same vector shifted to start from these different locations.

Step 2: Drawing the Vector from (0,0)

Start the vector \( \mathbf{u} \) at the origin \((0,0)\). The coordinates of the head of the vector will be \((-7, 2)\) because we add the vector components to the start point: \((0 + (-7), 0 + 2) = (-7, 2)\).

Step 3: Drawing the Vector from (2,3)

Place the tail of the vector at \((2,3)\). The head of the vector will be at:\((2 + (-7), 3 + 2) = (-5, 5)\). Draw the vector from \((2,3)\) to \((-5,5)\).

Step 4: Drawing the Vector from (-3,5)

Set the starting point at \((-3,5)\). Add the vector components to this point to find the head:\((-3 + (-7), 5 + 2) = (-10, 7)\). Draw the vector from \((-3,5)\) to \((-10,7)\).

Key Concepts

Vector Representation

Vectors are a powerful tool in mathematics, often used to represent quantities that have both a magnitude and a direction. A vector is commonly represented in component form, such as \( \mathbf{u} = \langle -7, 2 \rangle \). This notation gives us a clear picture of the vector's movement:

• "\(-7\)" indicates movement 7 units in the negative direction along the x-axis.
• "\(2\)" shows movement 2 units in the positive direction along the y-axis.

This representation offers a simple way to understand and utilize the geometric properties of vectors. Visualizing vectors as arrows provides insight into how they affect a space or coordinate plane. Each component describes how far and in what direction the arrow moves horizontally and vertically.

Initial Points

An important concept in vector mathematics is the idea of an initial point. The initial point is where the vector begins, or its tail. Changing the initial point effectively shifts the vector across the coordinate plane, while keeping its direction and magnitude.

• For example, when a vector like \( \mathbf{u} = \langle -7, 2 \rangle \) is placed at different initial points, its path remains the same, but its starting position changes.
• In the exercise, we use initial points like \((0, 0)\), \((2, 3)\), and \((-3, 5)\) to sketch the vector at various positions.

This aspect helps in understanding the versatility of vectors in mapping out paths and transformations in geometrical spaces. Different initial points can be used to simulate movement or translation, a useful property in physics and engineering.

Vector Addition

Vector addition is another essential concept when handling vectors. It allows us to determine the resulting vector when two vectors are combined. However, it is also used in a slightly different context in our exercise. In the given problem, vector addition helps in figuring out where the head of the vector will be when we know its initial point.

• We add the components of the vector \( \mathbf{u} = \langle -7, 2 \rangle \) to the coordinates of the initial points, such as adding \(-7\) and \(2\) to each point \((x, y)\).
• This process translates to \((x + (-7), y + 2)\), helping us find where the vector will end.

Through vector addition, students learn how to move vectors around the coordinate plane and discover how vectors interact when combined or translated.

Coordinate Geometry

Coordinate Geometry is the study of geometry using a coordinate system. It's a method that uses algebra to explore geometric properties. In context of vectors, it provides a powerful framework for calculating positions and transformations.

• By using coordinates, we can precisely specify both the initial points and the endpoints of vectors.
• This allows us to draw vectors accurately on the plane by calculating endpoints based on initial points and vector components.

Understanding and applying coordinate geometry principles helps students visualize how mathematical concepts translate into visual forms on a grid. It bridges the gap between algebraic expressions and geometric intuition, enabling deeper comprehension and application in various fields.