Problem 22 - Algebra and Trigonometry

Question

\(19-22\) . Sketch the given vector with initial point \((4,3),\) and find the terminal point. $$ \mathbf{u}=\langle- 8,-1\rangle $$

Step-by-Step Solution

Answer

The terminal point is \((-4, 2)\).

Step 1: Identify the Components of the Vector

The given vector \( \mathbf{u} = \langle -8, -1 \rangle \) has components \( -8 \) for the horizontal part and \( -1 \) for the vertical part. This means you will move 8 units to the left and 1 unit down from the initial point.

Step 2: Determine the Initial Point

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

Step 3: Calculate the Terminal Point

To find the terminal point, apply the vector components to the initial point. \( x \) component: \( 4 - 8 = -4 \); \( y \) component: \( 3 - 1 = 2 \). Thus, the terminal point is \( (-4, 2) \).

Step 4: Sketch the Vector

Start by plotting the initial point \( (4, 3) \) on your graph. From there, move 8 units to the left and 1 unit down to reach the terminal point \( (-4, 2) \). Draw an arrow from the initial point to the terminal point to represent the vector.

Key Concepts

Understanding the Initial Point

The initial point of a vector is essentially the starting point from which a vector originates. In vector notation, this is denoted as a pair of coordinates, usually represented as \(x_1, y_1\). In our given exercise, the initial point is \(4, 3\). This point signifies where you start your journey along the vector's path.
Imagine standing at the initial point as if it's your "home base." You are about to take a walk that follows the vector's direction and magnitude. This point is crucial because all calculations for locating the terminal point start here.
For any vector-related problem, always identify the initial point clearly, as this will help you apply the vector components accurately.

Calculating the Terminal Point

To understand a vector fully, you need to know where your journey ends, which is called the terminal point. This point is calculated by applying the vector's components to the initial point.
Think about the vector \(\mathbf{u} = \langle -8, -1 \rangle\), which tells you to move 8 units to the left and 1 unit down. You start at the initial point \(4, 3\).

• For the x-coordinate: Subtract 8 from 4, resulting in -4.
• For the y-coordinate: Subtract 1 from 3, resulting in 2.

Therefore, the terminal point is \(-4, 2\). By understanding how to apply vector components, you'll be able to determine the endpoint of any vector with ease.

Exploring Vector Components

Vector components define the direction and the distance you travel from the initial point. They are typically represented as \(\langle a, b \rangle\), where "a" is the horizontal component and "b" is the vertical component.
In our exercise, the vector \(\mathbf{u} = \langle -8, -1 \rangle\) has:

• Horizontal Component: -8, indicating 8 units to the left since it's negative.
• Vertical Component: -1, indicating 1 unit down due to the negative sign.

These components provide all the information needed to move from the initial to the terminal point. By understanding vector components, you'll have a clearer picture of how vectors work in real-world scenarios, such as navigation and physics. This grasp will enable you to master any vector problem effortlessly.