Problem 53 - Precalculus with Limits

Question

Writing a Linear Combination of Unit Vectors Exercises \(53-56\) , the initial and terminal points of a vector are given. Write the vector as a linear combination \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(-2,1)} & {(3,-2)}\end{array}\)

Step-by-Step Solution

Answer

The vector with initial point (-2,1) and terminal point (3,-2), when represented as a linear combination of the unit vectors, is written as \( \vec{v} = 5 \mathbf{i} - 3 \mathbf{j}\).

Step 1: Determination of the Vector

In order to determine the vector, subtract the initial point from the terminal point. Hence:\[(3- (-2), -2- 1)\]which simplifies to\[(3+2, -2-1)\]Therefore, the vector is\[(5, -3)\]

Step 2: Writing the Vector as a Linear Combination of Unit Vectors

The vector \((5, -3)\) can be represented as linear combination of the unit vectors \( \mathbf{i} = (1,0)\) and \( \mathbf{j} = (0,1)\) in \(R^2\). The formula in general form is \( \vec{v} = a \mathbf{i} + b \mathbf{j}\) where a and b are scalars. Substituting for a and b we get:\[ \vec{v} = 5 \mathbf{i} - 3 \mathbf{j}\]

Key Concepts

Vector Representation

Understanding vector representation is crucial for tackling many problems in physics and mathematics. A vector is a mathematical object that has both magnitude and direction, usually illustrated as an arrow. In a two-dimensional plane, any vector can be depicted in terms of its horizontal (x-axis) and vertical (y-axis) components. The notation \[\begin{equation} \vec{v} = (x, y) \end{equation}\] represents a vector with a horizontal component 'x' and a vertical component 'y'.

These components correspond to the movement from the initial point to the terminal point of the vector. In the given exercise, the vector's movement from \[\begin{equation} (-2,1) \end{equation}\] to \[\begin{equation} (3,-2) \end{equation}\] is captured by calculating the changes in the x and y coordinates which is the essence of representing a vector in two-dimensional space.

Unit Vectors i and j

In two-dimensional space, unit vectors play a fundamental role. The two basic unit vectors are \[\begin{equation} \mathbf{i} and \mathbf{j}. \end{equation}\] The vector \[\begin{equation} \mathbf{i} \end{equation}\] represents a one-unit length in the direction of the x-axis, while \[\begin{equation} \mathbf{j} \end{equation}\] is a one-unit length in the direction of the y-axis. These vectors serve as building blocks for any other vector in the plane, acting as the 'DNA' of vector representation. By combining multiples of \[\begin{equation} \mathbf{i} and \mathbf{j}, \end{equation}\] you can represent any vector in the plane. In essence, they form a foundation or basis for the vector space of the plane, much like the basis of a building.

Vector Components

Diving into the vector components, they are the projections of a vector along the axes of the coordinate system. When a vector is broken down into its components, it is dissected into a part that moves horizontally and another that moves vertically. In the exercise, the vector from initial to terminal point results in \[\begin{equation} (5, -3), \end{equation}\] which tells us that it moves 5 units in the direction of the x-axis, and -3 units in the direction of the y-axis.

Using the unit vectors \[\begin{equation} \mathbf{i} and \mathbf{j}, \end{equation}\] we scale them by these component values to create a linear combination. This method is a mathematical way to combine certain quantities scaling them accordingly. The result, \[\begin{equation} 5\mathbf{i} - 3\mathbf{j} \end{equation}\] is the vector \[\begin{equation} \vec{v} \end{equation}\] in terms of unit vectors—providing a clear and concise portrayal of its movement in two-dimensional space.