Problem 3 - Calculus: Early Transcendentals

Question

Let \(v\) be the vector with tail at the origin and head at (1,2)\(;\) let \(w\) be the vector with tail at the origin and head at \((3,1) .\) Draw \(v\) and \(w\) and a vector \(u\) with tail at (1,2) and head at \((3,1) .\) Draw \(\boldsymbol{u}\) with its tail at the origin.

Step-by-Step Solution

Answer

Vectors: \( v = (1, 2) \), \( w = (3, 1) \), \( u = (2, -1) \) at origin.

Step 1: Understand the Vectors

First, understand the vectors given. Vector \( v \) is from the origin \((0, 0)\) to \((1, 2)\). Vector \( w \) is from the origin \((0, 0)\) to \((3, 1)\).

Step 2: Calculate Vector u

Vector \( u \) starts at \((1, 2)\) and ends at \((3, 1)\). To find \( u \), subtract the coordinates of the start point from the end point:\[u = (3, 1) - (1, 2) = (2, -1)\]

Step 3: Translate Vector u to Origin

To draw vector \( u \) with its tail at the origin, use the calculated components:Tail at \((0, 0)\), head at \((2, -1)\). The new vector from the origin is \((2, -1)\).

Step 4: Draw the Vectors

Draw vector \( v \) from \((0, 0)\) to \((1, 2)\), vector \( w \) from \((0, 0)\) to \((3, 1)\), and vector \( u \) from the origin to \((2, -1)\). Make sure to clearly label each vector.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, plays a pivotal role in the analysis and visual representation of mathematical concepts. In coordinate geometry, every point on the plane is represented using a pair of numerical coordinates. These coordinates measure the horizontal and vertical distances from a predetermined point called the origin. The origin in a Cartesian coordinate system is denoted as \((0, 0)\).

For instance, consider the point \((1, 2)\). This indicates that the point is 1 unit along the x-axis (horizontal direction) and 2 units along the y-axis (vertical direction).

• The x-coordinate is the first number, which describes its position along the horizontal axis.
• The y-coordinate is the second number, representing the vertical position.

To understand vectors in coordinate geometry, it is crucial to grasp the concept of points because vectors frequently utilize these coordinate points to specify their starting or ending locations on the plane.

In our problem, vector \(v\) and vector \(w\) start from the origin, which is essential because vectors \(v\) and \(w\) can be seen as lines extending from the origin to the specified coordinates. Knowing how to accurately plot these points and vectors enables a clearer understanding of mathematical relationships in a visual context.

Vector Subtraction

Vector subtraction is a fundamental concept used to find the difference between two vectors by subtracting their corresponding components. This operation is essential in various mathematical tasks, including determining relative positions and transformations.

In the given exercise, vector subtraction is performed to find vector \(u\) that originates from point \((1, 2)\) and terminates at point \((3, 1)\). To calculate the components of vector \(u\), we subtract the coordinates of the tail (or starting point) from those of the head (or endpoint):

• The x-component of \(u\) is derived by subtracting the starting x-coordinate from the ending x-coordinate: \(3 - 1 = 2\).
• The y-component is found similarly: \(1 - 2 = -1\).

Thus, vector \(u\) can be described by the components \((2, -1)\). Understanding how to conduct vector subtraction allows us to determine not only directions and magnitudes but also the movement and spatial relations between different vectors.

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system. Every vector in a two-dimensional plane can be broken down into its horizontal (x-component) and vertical (y-component) components, providing a clear picture of the vector's direction and magnitude.

In the case of vector \(v\), starting at the origin and ending at \((1, 2)\), its components directly correspond to its endpoint coordinates.

• Its x-component is \(1\).
• Its y-component is \(2\).

For vector \(w\), its endpoint is at \((3, 1)\), so:

• x-component is \(3\).
• y-component is \(1\).

Breaking down vectors into components is a useful skill, especially when calculating resultant vectors through vector addition and subtraction. The components allow us to easily apply mathematical operations like these on vectors.

Understanding components is also critical in translating vectors. For instance, vector \(u\) with components \((2, -1)\) can be visualized from any point by applying its components along the respective axes to new starting points. In the exercise, this is illustrated when vector \(u\) is moved such that its tail is located at the origin.