Problem 22 - Precalculus Mathematics for Calculus

Question

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 of the vector is (-4, 2).

Step 1: Understand the Components of the Vector

The vector \( \mathbf{u} = \langle -8, -1 \rangle \) has components that tell us how to move from its initial point to find the terminal point. The first component \( -8 \) is a horizontal movement (left), and the second component \( -1 \) is a vertical movement (downwards).

Step 2: Analyze the Initial Point

The initial point of the vector is given as \( (4, 3) \). This is where the vector starts on the coordinate plane.

Step 3: Calculate the Terminal Point

Starting from the initial point \( (4, 3) \), apply the components of the vector: move 8 units to the left (subtract 8 from the x-coordinate) and 1 unit downwards (subtract 1 from the y-coordinate). This calculation is done as follows:- New x-coordinate: \( 4 + (-8) = -4 \)- New y-coordinate: \( 3 + (-1) = 2 \)Thus, the terminal point is \( (-4, 2) \).

Step 4: Sketch the Vector

On a coordinate plane, plot the initial point \( (4, 3) \). From this point, draw an arrow 8 units to the left and 1 unit downward to reach \( (-4, 2) \). This forms the directed line segment representing the vector \( \mathbf{u} \).

Key Concepts

Initial Point

In the realm of vector operations, the initial point serves as the anchor or starting position of a vector on a coordinate plane. When dealing with vectors, it's crucial to identify where the movement begins. An initial point is usually presented in terms of coordinates. These are the set of numbers that line up with specific points on the x-axis and y-axis.
For the given exercise, the initial point is clearly identified as \((4, 3)\). This means that before any movement along the vector's path is accounted for, you begin at this exact spot on the plane.

• The x-coordinate of 4 tells us how far to the right (positive) from the origin the point is.
• The y-coordinate of 3 tells us how high up (positive) from the x-axis the point is.

In essence, understanding and accurately plotting the initial point provides the fixed reference necessary for determining where the vector leads.

Terminal Point

The terminal point is where the journey of the vector concludes. After starting at the initial point, the vector moves according to its components and ultimately rests at the terminal point. This is the endpoint, encapsulating how far and in which direction the initial point has been displaced.
To determine the terminal point for the vector \( \mathbf{u} = \langle -8, -1 \rangle \) starting from \((4, 3)\), we apply the vector's components:

• Horizontal Movement: The x-component \(-8\) suggests a leftward movement of 8 units. To calculate: \(4 + (-8) = -4\).
• Vertical Movement: The y-component \(-1\) signifies a downward shift of 1 unit. To calculate: \(3 + (-1) = 2\).

Thus, the terminal point is exactly \((-4, 2)\). Knowing both the initial and terminal points not only helps in sketching the vector but also in understanding its impact and direction.

Coordinate Plane Movement

Vectors on a coordinate plane are essentially blueprints of movement described by their components. The coordinate plane is divided into quadrants and serves as a map for measuring shifts in direction and distance.
When analyzing movement through vectors:

• The x-component tells you about horizontal shifts. Positive values move right; negative values move left.
• The y-component describes vertical moves. Positive values ascend; negative values descend.

For the vector \(\mathbf{u} = \langle -8, -1 \rangle\), it communicates a movement of 8 units to the left and 1 unit downward from the starting point \((4, 3)\). This shifts your location across the different quadrants of the plane. Grasping this movement guides you in accurately portraying and knowing a vector's trajectory and endpoints. Proper comprehension of these movements is key to graphically and algebraically representing vectors in mathematical problems.