Problem 19 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The terminal point is \((6, 7)\).

Step 1: Understand the Problem

We need to sketch a vector \( \mathbf{u} = \langle 2, 4 \rangle \) starting from an initial point \((4,3)\) and find the terminal point of the vector.

Step 2: Determine the Direction and Magnitude of the Vector

The vector \( \mathbf{u} = \langle 2, 4 \rangle \) indicates that we move 2 units in the \(x\)-direction and 4 units in the \(y\)-direction from the initial point.

Step 3: Calculate the Terminal Point

Add the components of the vector \( \mathbf{u} \) to the initial point coordinates:\[ (4 + 2, 3 + 4) = (6, 7) \]Thus, the terminal point of the vector is \( (6, 7) \).

Step 4: Sketch the Vector

Plot the initial point \((4,3)\) on a coordinate plane. From this point, draw an arrow 2 units to the right and 4 units up to reach the terminal point \((6,7)\). This arrow represents the vector \( \mathbf{u} \).

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a two-dimensional surface that extends horizontally and vertically, defined by two perpendicular lines called axes. These axes intersect at the origin, marked as the point (0,0). The horizontal axis is known as the x-axis, while the vertical axis is referred to as the y-axis. Together, these axes divide the plane into four quadrants. Each point on this plane can be identified by a pair of coordinates, written in the form (x, y).

• The x-coordinate tells you the position to the left or right of the y-axis.
• The y-coordinate tells you the position above or below the x-axis.

In tasks involving vectors, like in our exercise, the coordinate plane helps visualize the starting and ending points of the vector. In our specific example, we plotted the initial point at (4, 3) and used the vector components to find the terminal point (6, 7). The process of graphing these points illustrates how vectors change position in the plane.

Exploring Vector Components

Vectors have two crucial components: magnitude and direction, both represented conveniently on the coordinate plane by the vector components. In this discussion, we use these components to convey the idea that vectors have an effect on position.

• The x-component shows the vector's influence along the horizontal axis. For example, our vector \( \mathbf{u} = \langle 2, 4 \rangle \) moves 2 units right in the x direction.
• The y-component indicates vertical movement. In \( \mathbf{u} \), it moves 4 units upward in the y direction.

This understanding of vector components allows us to navigate the coordinate plane more effectively. We determine overall movement, which involves moving a certain number of unit steps along each component's respective axis. The formula to find the terminal point by adding these components to the initial point coordinates simplifies this process.

Identifying Initial and Terminal Points

In vector operations, the initial point and terminal point are crucial to understanding a vector's start and end location. The initial point is where the vector begins, and the terminal point is where it ends, showing the vector's displacement.

• The initial point is given directly; in our exercise, it is (4, 3).
• The terminal point is found by applying the changes in direction and distance provided by the vector components to the initial point.

We add the respective components of the vector to the coordinates of the initial point to locate the terminal point. In our example, the vector \( \mathbf{u} = \langle 2, 4 \rangle \) starts at (4, 3) and ends at (6, 7) after adding the vector components to the initial point coordinates. This calculation effectively shows the path of movement and helps in graphically representing how a vector transforms the initial position to the final position on the coordinate plane.