Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Given vectors u and v, find (a) \(2 u\) (b) \(2 u+3 v\) (c) \(v-3 u\) Do not use a calculator. $$\mathbf{u}=2 \mathbf{i}, \mathbf{v}=\mathbf{i}+\mathbf{j}$$

Step-by-Step Solution

Answer

(a) \(4 \mathbf{i}\); (b) \(7 \mathbf{i} + 3 \mathbf{j}\); (c) \(-5 \mathbf{i} + \mathbf{j}\)."

Step 1: Calculate 2u

The vector \( \mathbf{u} \) is given as \( 2 \mathbf{i} \). To find \( 2 \mathbf{u} \), multiply each component of \( \mathbf{u} \) by 2. \[ 2 \mathbf{u} = 2 \times 2 \mathbf{i} = 4 \mathbf{i} \]. So, \( 2 \mathbf{u} = 4 \mathbf{i} \).

Step 2: Calculate 3v

The vector \( \mathbf{v} \) is given as \( \mathbf{i} + \mathbf{j} \). To find \( 3 \mathbf{v} \), multiply each component of \( \mathbf{v} \) by 3. \[ 3 \mathbf{v} = 3(\mathbf{i} + \mathbf{j}) = 3 \mathbf{i} + 3 \mathbf{j} \]. So, \( 3 \mathbf{v} = 3 \mathbf{i} + 3 \mathbf{j} \).

Step 3: Calculate 2u + 3v

Using the results from Steps 1 and 2, we find \( 2u + 3v \). Substitute \( 2u = 4i \) and \( 3v = 3i + 3j \). \[ 2u + 3v = 4\mathbf{i} + (3\mathbf{i} + 3\mathbf{j}) = (4\mathbf{i} + 3\mathbf{i}) + 3\mathbf{j} = 7\mathbf{i} + 3\mathbf{j} \].

Step 4: Calculate v - 3u

First, calculate \( 3u \) using vector \( \mathbf{u} = 2\mathbf{i} \). \[ 3\mathbf{u} = 3 \times 2\mathbf{i} = 6\mathbf{i} \]. Now, substitute \( \mathbf{v} = \mathbf{i} + \mathbf{j} \) and \( 3\mathbf{u} = 6\mathbf{i} \) into the expression. \[ v - 3u = (\mathbf{i} + \mathbf{j}) - 6\mathbf{i} = \mathbf{i} - 6\mathbf{i} + \mathbf{j} = -5\mathbf{i} + \mathbf{j} \].

Key Concepts

Scalar Multiplication

Scalar multiplication is one of the fundamental operations in vector algebra. It involves multiplying a vector by a scalar, which is a real number.
When you multiply a vector by a scalar, you scale or stretch its length by that scalar, without changing its direction, except when the scalar is negative, which reverses the direction.
For example, if we have a vector \( \mathbf{u} = 2 \mathbf{i} \), and we want to find \( 2 \mathbf{u} \), we multiply 2 (the scalar) by each component of \( \mathbf{u} \).

• \( 2 \mathbf{u} = 2 \times 2 \mathbf{i} = 4 \mathbf{i} \).

Thus, \( 2 \mathbf{u} \) is simply a vector that points in the same direction as \( \mathbf{u} \) but is twice as long. Scalar multiplication helps in adjusting the magnitude of vectors easily.

Vector Addition

Vector addition is a straightforward process where you add corresponding components of vectors together. This operation combines two or more vectors into a single resultant vector.
Given vectors, the rule is simple: add the \( \mathbf{i} \) components together and the \( \mathbf{j} \) components together.
For example, let's consider vectors \( 2 \mathbf{u} \) and \( 3 \mathbf{v} \):

• \( 2 \mathbf{u} = 4 \mathbf{i} \) since \( 2 \times 2 \mathbf{i} = 4 \mathbf{i} \).
• \( 3 \mathbf{v} = 3 \times (\mathbf{i} + \mathbf{j}) = 3 \mathbf{i} + 3 \mathbf{j} \).

To add these, simply sum their components:

• \( 2 \mathbf{u} + 3 \mathbf{v} = 4 \mathbf{i} + 3 \mathbf{i} + 3 \mathbf{j} = 7 \mathbf{i} + 3 \mathbf{j} \).

Hence, vector addition results in \( 7 \mathbf{i} + 3 \mathbf{j} \).
Note that the order in which you add vectors does not matter; the result will be the same, making vector addition commutative.

Vector Subtraction

Vector subtraction is similar to vector addition but involves finding the difference between vectors. This process can be thought of as adding a negative vector.
Subtracting vectors involves subtracting the components of one vector from the other. For example, with vectors \( \mathbf{v} = \mathbf{i} + \mathbf{j} \) and \( 3 \mathbf{u} = 6 \mathbf{i} \):

• Find the opposite (or negative) of \( 3 \mathbf{u} \) as \( -6 \mathbf{i} \).

Then, to compute \( \mathbf{v} - 3 \mathbf{u} \):

• \( (\mathbf{i} + \mathbf{j}) - 6 \mathbf{i} = \mathbf{i} - 6 \mathbf{i} + \mathbf{j} \).
• This simplifies to \( -5 \mathbf{i} + \mathbf{j} \).

Thus, vector subtraction yields the vector \( -5 \mathbf{i} + \mathbf{j} \).
Understanding vector subtraction is crucial for determining relative positions and motions in physics and engineering.

Unit Vectors

Unit vectors are vectors with a magnitude of one. They are crucial for defining directions and are commonly used in vector operations to simplify calculations and express vectors in standard forms.
In the context of Cartesian coordinates, the typical unit vectors are \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).

• \( \mathbf{i} \) is a unit vector pointing in the x-direction.
• \( \mathbf{j} \) is a unit vector pointing in the y-direction.

These unit vectors help represent other vectors by scaling them. For instance, a vector like \( \mathbf{v} = \mathbf{i} + \mathbf{j} \) can be viewed as having components in both the x and y directions, each represented by the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \).
Unit vectors serve as essential building blocks in vector mathematics, serving to define directions and help construct vectors from scalar multiples.