Problem 20 - Algebra and Trigonometry

Question

\(19-22\) . Sketch the given vector with initial point \((4,3),\) and find the terminal point. $$ \mathbf{u}=\langle- 1,2\rangle $$

Step-by-Step Solution

Answer

The terminal point is (3, 5).

Step 1: Understand the Vector Components

The vector \( \mathbf{u} = \langle -1, 2 \rangle \) tells us how to move from the initial point. The x-component, \(-1\), indicates a movement of 1 unit to the left, and the y-component, \(2\), indicates a movement of 2 units up.

Step 2: Identify the Initial Point

The initial point of the vector is given as \((4, 3)\). This is where the tail of the vector is located.

Step 3: Calculate the Terminal Point

To find the terminal point, apply the vector's components to the initial point:- For the x-coordinate: \(4 - 1 = 3\)- For the y-coordinate: \(3 + 2 = 5\)Thus, the terminal point is \((3, 5)\).

Step 4: Sketch the Vector

Start by plotting the initial point \((4, 3)\) on the coordinate plane. From this point, move 1 unit left and 2 units up to reach the terminal point \((3, 5)\). Draw a directed line segment (arrow) from \((4, 3)\) to \((3, 5)\) to represent the vector \( \mathbf{u} \).

Key Concepts

Terminal Point

The terminal point of a vector is the endpoint, reached after applying the vector components to an initial point. Think of it as the end of a journey, guided by the directions provided by the vector. In the case of the vector \( \mathbf{u} = \langle -1, 2 \rangle \), the initial point is \((4, 3)\). Moving from this initial point, we apply the components as follows:

• The negative x-component \(-1\) means we move 1 unit to the left.
• The positive y-component \(2\) means we move 2 units upward.

Starting from \((4, 3)\), we subtract 1 from the x-coordinate and add 2 to the y-coordinate, resulting in the terminal point \((3, 5)\). This point is where the head of the vector arrives, completing its path from the initial point.

Initial Point

Every vector has a starting location called the initial point. It's where you place the tail of the vector before it "travels" via its components. In our exercise, the initial point is \((4, 3)\). This is often provided in exercises and is crucial for positioning the vector on a graph.

• You start at this point before applying the vector's directions.
• It's important to plot this point correctly to ensure the vector's path is accurate.

The initial point serves as a reference, making it easy to apply the vector's changes in the x and y coordinates. Visualizing this can help you understand the vector's overall movement, stemming directly from this starting position.

Vector Components

Vector components are the elements that dictate the vector’s direction and magnitude. They help in pinpointing precisely how the vector moves in each coordinate axis. A vector like \( \mathbf{u} = \langle -1, 2 \rangle \) is broken down as follows:

• \(-1\) is the x-component, indicating a horizontal move to the left.
• \(2\) is the y-component, indicating a vertical move upwards.
• The components define the vector's geometry and facilitate sketching it on a coordinate plane.

Understanding these components is key to predicting the vector's endpoint, or terminal point, when added to its initial point. They also allow you to sketch vectors accurately, giving a clear visual representation of abstract mathematical concepts on a 2D plane.