Problem 20 - Precalculus Mathematics for Calculus

Question

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: Understanding the Vector Problem

We have a vector \( \mathbf{u} = \langle -1, 2 \rangle \). This vector represents a direction and magnitude. When applied to an initial point \((4, 3)\), it will allow us to find the terminal point of the vector.

Step 2: Applying the Vector to the Initial Point

To find the terminal point, add the components of the vector \( \mathbf{u} \) to the initial point \((4, 3)\). The vector adds \(-1\) to the x-coordinate and \(2\) to the y-coordinate. So, the calculations are: \(4 + (-1) = 3\) for the x-coordinate and \(3 + 2 = 5\) for the y-coordinate.

Step 3: Determining the Terminal Point

By applying the vector \( \mathbf{u} \) to the initial point \((4, 3)\), the terminal point is found to be \((3, 5)\).

Step 4: Sketching the Vector

To sketch the vector, plot the initial point \((4, 3)\) on a coordinate plane. Draw an arrow from this point to the terminal point \((3, 5)\). The direction of the arrow represents the vector \( \mathbf{u} \).

Key Concepts

Vector Addition

Vector addition is essential in understanding how to manipulate vectors in a two-dimensional or three-dimensional space. Vectors are described by their components, typically noted in angle brackets, such as \( \mathbf{u} = \langle -1, 2 \rangle \). Each component corresponds to a movement along an axis in the coordinate plane.

To perform vector addition when given an initial point, simply add each corresponding component of the vector to the initial point's coordinates. Given our initial point \((4, 3)\) and the vector \( \mathbf{u} \), you add \(-1\) to the x-coordinate and \(2\) to the y-coordinate. Therefore:

• \(x : 4 + (-1) = 3\)
• \(y : 3 + 2 = 5\)

This calculation results in the terminal point \((3, 5)\) for the vector. It's a straightforward process that, once understood, allows one to easily compute the resulting locations of translated points in the plane.

Coordinate Geometry

Coordinate geometry, also called analytic geometry, is the study of geometry using a coordinate system. It's the tool we use to locate points in space using ordered pairs. For our example, we start with the initial point \((4, 3)\) and apply vector addition to find the new location.

The coordinate plane has two axes: the x-axis (horizontal) and the y-axis (vertical). Points on this plane are defined by their x (horizontal) and y (vertical) values. When doing vector addition, each component of the vector shifts the position within this plane.

Vectors like \( \mathbf{u} = \langle -1, 2 \rangle \) dictate how far and in which direction we move from the initial point. The vector's influence changes the location along these axes, i.e., moving left by 1 unit and upwards by 2 units from \((4, 3)\) to \((3, 5)\).

This method of graphing and calculating with coordinates allows us to visualize and solve geometric problems accurately.

Vector Sketching

To effectively communicate the result of vector mathematics, sketching is invaluable. Vector sketching on a coordinate plane involves illustrating the direction and magnitude of the vector clearly.

Start by plotting the initial point. In this example, plot \((4, 3)\) on the graph. The vector \( \mathbf{u} = \langle -1, 2 \rangle \) then directs us to the terminal point \((3, 5)\). Draw an arrow from the initial point to the terminal point.

The length of the arrow indicates the vector's magnitude, while the arrowhead points in the vector's specific direction. This visual representation helps in understanding how data points shift in a given space due to vector application.

Through sketching, one can better grasp the idea of directional movement and the resultant position, conveying complex calculations as intuitive visuals.