Problem 25 - Precalculus Mathematics for Calculus

Question

Sketch representations of the given vector with initial points at \((0,0),(2,3),\) and \((-3,5)\) $$\mathbf{u}=\langle- 7,2\rangle$$

Step-by-Step Solution

Answer

Sketch vectors from each initial point to the resulting endpoints: (-7,2), (-5,5), and (-10,7).

Step 1: Understand the Vector Representation

The vector \(\mathbf{u}\) is given as \(\langle -7, 2 \rangle\). This means the vector has a horizontal component of -7 and a vertical component of 2. It can be represented as moving 7 units to the left and 2 units up from its initial point.

Step 2: Sketch Vector from (0,0)

Start at the origin, (0,0). From here, move 7 units to the left to the point (-7,0) and then 2 units upwards to reach (-7,2). Draw an arrow from (0,0) to (-7,2) to represent the vector \(\mathbf{u}\).

Step 3: Sketch Vector from (2,3)

Starting at the point (2,3), move 7 units left to reach (-5,3) and then 2 units up to arrive at (-5,5). Draw an arrow from (2,3) to (-5,5) to represent the vector \(\mathbf{u}\).

Step 4: Sketch Vector from (-3,5)

Starting from the point (-3,5), move 7 units to the left to (-10,5), and then 2 units upward to (-10,7). Draw an arrow from (-3,5) to (-10,7) to represent the vector \(\mathbf{u}\).

Key Concepts

Sketching Vectors

Vectors are mathematical objects that have both a direction and a magnitude. When sketching vectors, the process involves drawing arrows that visualize these two properties.
To effectively sketch a vector, begin by identifying the initial point, which serves as the starting point of your arrow. Then, consider the vector's components, typically given in terms of horizontal and vertical shifts.

• The horizontal component indicates how far and in which direction (left or right) you should move from the initial point.
• The vertical component tells you how far and in which direction (up or down) to move afterward.

Place the terminal point where these components land you, and draw a straight arrow from the initial to the terminal point. This arrow represents your vector. Making these sketches helps visualize and understand vectors better, especially when dealing with various initial points.

Vector Components

The components of a vector play a crucial role in defining its direction and length. Typically written in the format \( \langle x, y \rangle \), these components indicate:

• \( x \) - the horizontal shift from the initial point: a negative value means move left, while a positive value means move right.
• \( y \) - the vertical shift: a positive value indicates a move upwards, while a negative one points downwards.

For the vector \( \mathbf{u} = \langle -7, 2 \rangle \), this translation involves moving 7 units to the left and 2 units upward.
It's essential to visualize these movements accurately to determine where the terminal point lies. The vector's components offer a structured way to navigate the coordinate plane effectively.

Initial Points and Terminal Points

Understanding the positions of initial and terminal points is vital when working with vectors. The initial point, where you start drawing the vector, can be any given coordinate.

• For example, with initial points like \((0,0), (2,3),\) and \((-3,5)\), each requires a unique approach to drawing the vector.

From each initial point, apply the vector's components to find the terminal point. This involves shifting horizontally and vertically according to the vector's description.

• For \((0,0)\), moving to \((-7,2)\) positions the terminal point.
• From \((2,3)\), \((-5,5)\) becomes the terminal point.
• With \((-3,5)\), the terminal point is \((-10,7)\).

Drawing a line between initial and terminal points with an arrow going toward the terminal point visually represents the vector. This method clarifies how vectors function in different positions on a plane.