Problem 23 - Algebra and Trigonometry

Question

\(23-26\) Sketch representations of the given vector with initial points at \((0,0),(2,3),\) and \((-3,5)\). $$ \mathbf{u}=\langle 3,5\rangle $$

Step-by-Step Solution

Answer

Sketch the vector \(\mathbf{u} = \langle 3, 5 \rangle\) starting from the points \((0,0)\), \((2,3)\), and \((-3,5)\), ending at \((3,5)\), \((5,8)\), and \((0,10)\) respectively.

Step 1: Understand the vector definition

The vector \( \mathbf{u} = \langle 3, 5 \rangle \) indicates a vector that has a horizontal component of 3 units and a vertical component of 5 units.

Step 2: First initial point (0,0)

Start at the origin \((0, 0)\). From the origin, move 3 units to the right and 5 units up. The endpoint will be \((3, 5)\). Draw an arrow from \((0, 0)\) to \((3, 5)\) to represent the vector \( \mathbf{u} \).

Step 3: Second initial point (2,3)

Begin at the point \((2, 3)\). From this point, move 3 units to the right and 5 units up. The endpoint will be \((5, 8)\). Draw an arrow from \((2, 3)\) to \((5, 8)\) for vector \( \mathbf{u} \).

Step 4: Third initial point (-3,5)

Starting at \((-3, 5)\), move 3 units to the right and 5 units up. The endpoint will be \((0, 10)\). Draw an arrow from \((-3, 5)\) to \((0, 10)\) for the representation of the vector \( \mathbf{u} \).

Key Concepts

Vector Components

In the world of vectors, each vector can be broken down into its components. These components are essentially the building blocks that define the vector. For the given vector \( \mathbf{u} = \langle 3, 5 \rangle \), there are two components:

• The *horizontal* component which is 3.
• The *vertical* component which is 5.

Understanding these components is crucial because they tell us how far and in what direction the vector moves along each axis. In this case, "3 units along the x-axis" means the vector moves 3 units to the right, while "5 units along the y-axis" dictates moving 5 units upwards. Together, \((3, 5)\) describe the vector's direction and magnitude in a 2D plane.

Graphical Representation

Visualizing vectors is an important skill, especially in understanding their application in physics and engineering. A vector, like \( \mathbf{u} = \langle 3, 5 \rangle \), is represented graphically as an arrow. The arrow illustrates direction and magnitude, originating from an initial point and terminating at an endpoint.Here's how you can visualize this:

• Start by plotting the initial point, which serves as the tail of the arrow.
• The direction of the arrow is established by moving parallel to the defined vector components: 3 steps horizontally and 5 steps vertically.
• The endpoint, where the arrowhead lands, marks where these component moves conclude.

Using this graphical approach provides a clear, visual perspective of how vectors operate in a coordinate system.

Initial Point

The initial point of a vector is the starting location on a graph from which the vector begins its journey. It's a significant concept because it establishes the basis for vector transformations. For instance:

• When the initial point is at \((0,0)\), it's termed as starting from the "origin". Here, the vector \( \mathbf{u} \) simply ends at \((3, 5)\), making it straightforward to map.
• If the vector begins at \((2, 3)\), you're essentially shifting the entire path of the vector further along the plane to end at \((5, 8)\).
• Starting from \((-3, 5)\) involves a translation where, despite the initial point shift, the vector maintains its set direction and magnitude, terminating at \((0, 10)\).

Thus, changing the initial point only shifts the vector's path, but not its intrinsic properties like direction and magnitude.

Endpoint of Vector

The endpoint is the final destination of a vector once it has traveled its course from the initial point through its components. It is essential in defining the direction and orientation of the vector.With \( \mathbf{u} = \langle 3, 5 \rangle \), no matter where the vector starts, taking 3 steps right and 5 steps up will always direct it to a unique endpoint:

• From \((0,0)\) the endpoint is \((3,5)\).
• Starting at \((2,3)\) leads to \((5,8)\).
• Beginning from \((-3,5)\) concludes at \((0,10)\).

This transformation displays how a vector's endpoint depends both on its initial point and its components. The endpoint precisely locates the vector once its movements are completed, signaling its full representation on a plane.