Problem 35 - Precalculus with Limits

Question

Vector Operations In Exercises 31-38, find (a) \(\mathbf{u}+\mathbf{v}\) . (b) \(\mathbf{u}-\mathbf{v},\) and \((\mathbf{c}) 2 \mathbf{u}-3 \mathbf{v} .\) Then sketch each resultant vector. $$\mathbf{u}=\mathbf{i}+\mathbf{j}, \mathbf{v}=2 \mathbf{i}-3 \mathbf{j}$$

Step-by-Step Solution

Answer

The results of the operations are \((a) \mathbf{u}+\mathbf{v} = 3\mathbf{i}-2\mathbf{j}\), \((b) \mathbf{u}-\mathbf{v} = -\mathbf{i} + 4\mathbf{j}\), \((c) 2\mathbf{u} - 3\mathbf{v} = -4\mathbf{i} + 11 \mathbf{j}\). The graphical representations of these vectors would represent these calculations on a coordinate plane.

Step 1: Vector Addition

To add the vectors \( \mathbf{u} \) and \( \mathbf{v} \), add the corresponding \( \mathbf{i} \) and \( \mathbf{j} \) components together. For \( \mathbf{u}+\mathbf{v} \), this means adding \( \mathbf{i} + 2\mathbf{i} \) and \( \mathbf{j} - 3\mathbf{j} \), which yields \( 3i - 2j \).

Step 2: Vector Subtraction

To subtract the vectors \( \mathbf{u} \) and \( \mathbf{v} \), subtract the corresponding \( \mathbf{i} \) and \( \mathbf{j} \) components. For \( \mathbf{u}-\mathbf{v} \), this means subtracting \( 2\mathbf{i} \) from \( \mathbf{i} \) and \( -3\mathbf{j} \) from \( \mathbf{j} \), which yields \( -\mathbf{i} + 4\mathbf{j} \).

Step 3: Scalar multiplication and Vector Subtraction

To find \( 2\mathbf{u} - 3\mathbf{v}\), multiply the vector \( \mathbf{u} \) by 2 and \( \mathbf{v} \) by 3, then subtract the resulting vectors according to the rules of vector subtraction. This gives you \( 2(\mathbf{i}+\mathbf{j}) - 3(2\mathbf{i}-3\mathbf{j}) = -4\mathbf{i} + 11 \mathbf{j} \).

Step 4: Vector Sketching

The vectors are sketched in a coordinate system with \( \mathbf{i} \) along the x-axis and \( \mathbf{j} \) along the y-axis. The vector \( \mathbf{i}+\mathbf{j} \) is a diagonal line in the first quadrant, \( 2\mathbf{i}-3\mathbf{j} \) is a line in the fourth quadrant, and their resultants (\( 3\mathbf{i}-2\mathbf{j} \), \( -\mathbf{i}+4\mathbf{j} \), \( -4\mathbf{i}+11\mathbf{j} \) respectively) follow the rules of vector addition and subtraction respectively.

Key Concepts

Vector Addition

Vector addition is a fundamental operation where two vectors are combined to form a new vector, called the resultant vetor. Imagine vectors as arrows. The tail of the "arrows" can be placed on a common point, and to add the vectors, you join the head of the first vector to the tail of the second. In terms of mathematics, you add their corresponding components.
For example, if we have vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - 3\mathbf{j} \), vector addition involves:

• Adding the \( \mathbf{i} \) components: \( \mathbf{i} + 2\mathbf{i} = 3\mathbf{i} \)
• Adding the \( \mathbf{j} \) components: \( \mathbf{j} - 3\mathbf{j} = -2\mathbf{j} \)

This results in the new vector: \( \mathbf{u} + \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \), indicating the cumulative effect of moving in the direction of each vector.

Vector Subtraction

Vector subtraction is just as crucial as vector addition. It can also be thought of as adding a negative version of a vector to another vector. Essentially, you are finding out how one vector differs from another in terms of their direction and magnitude.
To subtract the vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - 3\mathbf{j} \), follow the steps below:

• Subtract the \( \mathbf{i} \) components: \( \mathbf{i} - 2\mathbf{i} = -\mathbf{i} \)
• Subtract the \( \mathbf{j} \) components: \( \mathbf{j} - (-3\mathbf{j}) = \mathbf{j} + 3\mathbf{j} = 4\mathbf{j} \)

These calculations give the resultant vector as \( \mathbf{u} - \mathbf{v} = -\mathbf{i} + 4\mathbf{j} \). This expresses how the directional movement given by \( \mathbf{u} \) alters when "backtracking" by \( \mathbf{v} \).

Scalar Multiplication

Scalar multiplication involves scaling a vector by multiplying it with a scalar (a real number). This operation affects the vector's magnitude, stretching or shrinking it, while maintaining its direction if the scalar is positive.
In the case of vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - 3\mathbf{j} \), performing scalar multiplication for the expression \( 2\mathbf{u} - 3\mathbf{v} \) involves:

• Multiplying \( \mathbf{u} \) by 2: results in \( 2\mathbf{i} + 2\mathbf{j} \)
• Multiplying \( \mathbf{v} \) by 3: results in \( 6\mathbf{i} - 9\mathbf{j} \)

Next, employ vector subtraction, \( 2\mathbf{u} - 3\mathbf{v} = (2\mathbf{i} + 2\mathbf{j}) - (6\mathbf{i} - 9\mathbf{j}) \) which simplifies to:
Subtract \( 6\mathbf{i} \) from \( 2\mathbf{i} \), and add \( 9\mathbf{j} \) to \( 2\mathbf{j} \):

• \( 2\mathbf{i} - 6\mathbf{i} = -4\mathbf{i} \)
• \( 2\mathbf{j} + 9\mathbf{j} = 11\mathbf{j} \)

The final resultant vector is \( -4\mathbf{i} + 11\mathbf{j} \), expressing the combined effect of scaling \( \mathbf{u} \) and \( \mathbf{v} \), and then performing subtraction.