Problem 21 - Algebra and Trigonometry

Question

\(19-22\) . Sketch the given vector with initial point \((4,3),\) and find the terminal point. $$ \mathbf{u}=\langle 4,-3\rangle $$

Step-by-Step Solution

Answer

Terminal point is (8, 0).

Step 1: Identify the Initial Point

The initial point of the vector is given as \((4, 3)\). This is where the vector will start on the graph.

Step 2: Understand the Vector Components

The vector \( \mathbf{u} = \langle 4, -3 \rangle \) means that the vector moves 4 units to the right and 3 units down from the initial point.

Step 3: Calculate the Terminal Point

To find the terminal point, add the vector components to the initial point:- For the x-coordinate: Add 4 to the initial x-value: \(4 + 4 = 8\)- For the y-coordinate: Subtract 3 from the initial y-value: \(3 - 3 = 0\)So, the terminal point is \((8, 0)\).

Step 4: Sketch the Vector

Draw the vector on a coordinate plane starting at \((4, 3)\) and ending at \((8, 0)\). The vector \( \mathbf{u} \) directs 4 units to the right and 3 units down.

Key Concepts

Initial Point and Terminal Point

When working with vectors, it's crucial to understand the concepts of the initial point and terminal point. The initial point is where a vector begins in coordinate geometry.
For example, if a vector starts at point (4, 3), this is considered the initial point.
The terminal point, on the other hand, is where the vector ends after moving in the direction and magnitude specified by its components.

• Initial Point: The starting location, given as coordinates.
• Terminal Point: The endpoint, found by adding the vector components to the initial point's coordinates.

Understanding these points helps in sketching and analyzing vectors on a graph. Given the initial point (4, 3) and vector components explained in the exercise, the terminal point was calculated to be (8, 0).
By establishing clear start and finish points for the vector, we can better understand its direction and magnitude.

Vector Components

Vectors have components that define their direction and length. These components are typically written in angle bracket notation, such as \( \mathbf{u} = \langle 4, -3 \rangle \).
Components tell us how far and in which direction the vector moves from its initial point.
Let's break down the components:

• The first number (4) indicates the horizontal movement, which is to the right on a coordinate plane.
• The second number (-3) indicates the vertical movement, which is downward.

Each component affects how the vector translates from its starting point:- An increase in the first component signifies more rightward movement.- A decrease in the second component suggests downward movement.Using the components of a vector, we can easily determine where the vector moves from its starting point and where it ultimately arrives (the terminal point). In this case, from (4, 3) to (8, 0).

Graphing Vectors

Graphing vectors on a coordinate plane allows us to visually interpret their magnitude and direction.
The process involves plotting the initial point and then using the vector components to find and plot the terminal point.
In our exercise, graphing the vector \( \mathbf{u} = \langle 4, -3 \rangle \) follows these steps:

• Start by marking the initial point (4, 3) on the graph.
• Next, apply the vector components by moving 4 units to the right and 3 units down.
• Mark the terminal point (8, 0).
• Draw a line with an arrow from the initial point to the terminal point; the arrow indicates the direction of the vector.

This line represents the vector, showing both the path taken and the distance covered.
By seeing vectors drawn out, we gain insight into both their relative direction and scale, making them easier to analyze for further mathematical applications.