Problem 55 - 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 of the standard unit vectors i and j. \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(-6,4)} & {(0,1)}\end{array}\)

Step-by-Step Solution

Answer

The vector with initial point (-6,4) and terminal point (0,1) can be represented as a linear combination of the unit vectors i and j as \(6i - 3j\).

Step 1: Calculate Displacement in X direction

Displacement in the x direction is calculated by the formula \(Δx = x_{2} - x_{1}\) where \(x_{2}\) and \(x_{1}\) are the x-coordinates of the terminal and initial points respectively. Here, \(x_{2} = 0\) and \(x_{1} = -6\). Therefore, \(Δx = 0 - (-6) = 6\).

Step 2: Calculate the displacement in Y direction

Displacement in the y direction is calculated by the formula \(Δy = y_{2} - y_{1}\) where \(y_{2}\) and \(y_{1}\) are the y-coordinates of the terminal and initial points respectively. Here, \(y_{2} = 1\) and \(y_{1} = 4\). Therefore, \(Δy = 1 - 4 = -3\)

Step 3: Write the Vector as a Linear Combination of the Unit Vectors

The vector displacement can be represented as a linear combination of two unit vectors i and j. i is the unit vector in the x direction, and j is the unit vector in the y direction. Therfore, the vector based on the displacements in x and y directions can be represented as: \(6i - 3j\).

Key Concepts

Unit Vectors

Understanding unit vectors is like understanding the basic building blocks of vectors. Unit vectors are vectors with a magnitude of 1 and they indicate direction. They are incredibly useful because they help in breaking down vectors into their components.

• i: This unit vector points in the direction of the positive x-axis.
• j: This unit vector points in the direction of the positive y-axis.

Using these unit vectors, any vector in a two-dimensional plane can be expressed as a combination of these two. This means that you can add scaled i and j vectors to represent any vector's direction and length in the plane. This becomes especially clear when representing vectors as linear combinations.

Vector Representation

To represent a vector as a linear combination of unit vectors, you need to consider its impact in both the x and y direction. When you start with the initial and terminal points of a vector, you are essentially given the starting and ending locations of the movement.
From the exercise, you have:

• Initial Point: (-6, 4)
• Terminal Point: (0, 1)

The movement or change from the initial to terminal point can be expressed using unit vectors. If a vector has an x-component of 6 and a y-component of -3, we can write this as 6i - 3j, where 6 is the coefficient for the i component, and -3 is the coefficient for the j component. This represents how far the point moves along the x and y axes, respectively.

Displacement Calculation

Displacement calculation is a fundamental concept: it tells you how far and in which direction a point moves from a starting point. This is found using the coordinates of the initial and terminal points.
To find the displacement:

• X Displacement: Use the formula \( \Delta x = x_{2} - x_{1} \), where \( x_{2} \) is the x-term of the terminal point, and \( x_{1} \) is the x-term of the initial point. In this exercise, \( \Delta x = 0 - (-6) = 6 \).


• Y Displacement: Apply a similar method with\( \Delta y = y_{2} - y_{1} \), where \( y_{2} \) is the y-coordinate of the terminal point, and \( y_{1} \) is the initial.\( \Delta y = 1 - 4 = -3 \).

This step-by-step calculation helps in accurately determining the final representation of the vector using unit vectors i and j, where each step builds the complete picture of the vector's movement in the plane.