Problem 10 - Precalculus : Building Concepts and Connections

Question

Write each of the given vectors in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$\mathbf{v}=\langle-3.5,4\rangle$$

Step-by-Step Solution

Answer

The vector \(\mathbf{v} = \<-3.5,4>\) can be written in terms of the unit vectors as \(-3.5\mathbf{i} + 4\mathbf{j}\).

Step 1: Identify the vector components

The given vector \(\mathbf{v} = \<-3.5,4>\) has two components: its magnitude in the horizontal direction, which is -3.5, and its magnitude in the vertical direction, which is 4.

Step 2: Multiply the components by the unit vectors

Multiply each component of the vector by the corresponding unit vector. Thus the horizontal component -3.5 is multiplied by the unit vector in the horizontal direction \( \mathbf{i} \), and the vertical component 4 is multiplied by the unit vector in the vertical direction \( \mathbf{j} \). This results in \(-3.5\mathbf{i} + 4\mathbf{j}\).

Key Concepts

Vector Components

Understanding vector components is crucial in analyzing the direction and magnitude of vectors in a plane. Break a vector down into its horizontal and vertical parts, and these parts are called the vector's components. For instance, if you have a vector that moves leftwards and upwards from the origin, its horizontal movement is captured by one component, and its vertical movement is captured by another.

Specifically, when you look at the vector \(\textbf{v} = \<-3.5,4>\), the first value, -3.5, represents the horizontal component, meaning the vector moves 3.5 units left from the origin, as a negative sign indicates direction opposite to the positive x-axis. The second value, 4, is the vertical component, which tells us the vector moves 4 units up, as it's positive. Clarifying the concept of vector components aids in visualizing vector movements and conducting operations such as addition, subtraction, and scalar multiplication.

Unit Vector Notation

Unit vector notation is a standardized way to represent the direction of a vector along the axes of a coordinate system without regard to its magnitude. A unit vector is a vector that has a magnitude of one. In the two-dimensional Cartesian coordinate system, two perpendicular unit vectors are commonly used: \(\textbf{i}\) for the horizontal (x-axis) and \(\textbf{j}\) for the vertical (y-axis).

When a vector's components are multiplied by these unit vectors, they maintain their magnitude but take on the direction of the unit vectors. For example, the vector \(\textbf{v} = \<-3.5,4>\) when expressed in unit vector notation becomes \(-3.5\textbf{i} + 4\textbf{j}\). This form provides a direct expression of the vector's direction and magnitude with respect to the standard axes, making computations in vector algebra more systematic and comprehensible.

Vector Representation

Vector representation refers to the various ways in which vectors can be described mathematically. Two common forms of representation are component form and unit vector notation. In component form, a vector in two dimensions is written simply as a pair of numbers, \(\langle a, b \rangle\), signifying its horizontal and vertical movements from the origin.

However, when expressed in unit vector notation as seen in the earlier example \(-3.5\textbf{i} + 4\textbf{j}\), a vector's representation incorporates the unit vectors \(\textbf{i}\) and \(\textbf{j}\), providing clear directions along the x and y axes, respectively.

Why It Matters

This manner of expressing vectors is vital in fields like physics and engineering where precise vector direction and magnitude are essential. By understanding different representations of vectors, students can easily transition between abstract problems and real-world applications, as well as conduct operations like vector addition, scalar multiplication, or finding the magnitude and direction of vectors with confidence.