Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Given \(\mathbf{u}=\langle- 2,5\rangle\) and \(\mathbf{v}=\langle 4,3\rangle,\) find each vector. Do not use a calculator. $$3 u+6 v$$

Step-by-Step Solution

Answer

The vector is \( \langle 18, 33 \rangle \).

Step 1: Understand the Problem

We need to find the vector resulting from the linear combination of two vectors \(\mathbf{u}\) and \(\mathbf{v}\), specifically the expression \(3\mathbf{u} + 6\mathbf{v}\). This involves scalar multiplication of vectors and then vector addition.

Step 2: Scalar Multiplication of Vector \(\mathbf{u}\)

Start by multiplying each component of \(\mathbf{u} = \langle -2, 5 \rangle\) by 3. This results in the vector:\[ 3\mathbf{u} = 3 \langle -2, 5 \rangle = \langle 3 \times (-2), 3 \times 5 \rangle = \langle -6, 15 \rangle. \]

Step 3: Scalar Multiplication of Vector \(\mathbf{v}\)

Next, multiply each component of \(\mathbf{v} = \langle 4, 3 \rangle\) by 6. This results in the vector:\[ 6\mathbf{v} = 6 \langle 4, 3 \rangle = \langle 6 \times 4, 6 \times 3 \rangle = \langle 24, 18 \rangle. \]

Step 4: Add the Resulting Vectors

Now add the two vectors obtained from the previous steps: \(3\mathbf{u}\) and \(6\mathbf{v}\). Add the corresponding components of \(\langle -6, 15 \rangle\) and \(\langle 24, 18 \rangle\). This results in:\[ 3\mathbf{u} + 6\mathbf{v} = \langle -6 + 24, 15 + 18 \rangle = \langle 18, 33 \rangle. \]

Key Concepts

Understanding Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar value. This operation scales the vector. For example, if we multiply a vector by 3, each component of the vector is multiplied by 3.
The result is a new vector pointing in the same direction (if the scalar is positive) or the opposite direction (if negative).

• Consider the vector \(\mathbf{u} = \langle -2, 5 \rangle\).
• Multiply by 3: \(3\mathbf{u} = \langle 3 \times (-2), 3 \times 5 \rangle = \langle -6, 15 \rangle\).

Scalar multiplication affects both the length of the vector (makes it longer or shorter) and the direction (if the scalar is negative). This simple operation is often a first step in more complex calculations with vectors.

Exploring Linear Combination of Vectors

A linear combination of vectors involves adding scaled vectors together. This helps in finding a resultant vector that could represent a net force or movement.
To form a linear combination:

• Multiply each vector by a specific scalar.
• Add the resulting vectors together by adding corresponding components.

For example, given the vectors \(\mathbf{u} = \langle -2, 5 \rangle\) and \(\mathbf{v} = \langle 4, 3 \rangle\), and the expression \(3\mathbf{u} + 6\mathbf{v}\):

• We first perform scalar multiplication to get \(\langle -6, 15 \rangle\) from \(\mathbf{u}\) and \(\langle 24, 18 \rangle\) from \(\mathbf{v}\).
• Then, we add these two vectors to get the final vector \(\langle 18, 33 \rangle\).

This process is vital in various applications, such as physics, where different forces or movements combine into a single effect.

Decoding Vector Components

Vector components are the individual numbers that make up a vector, usually in two or three dimensions. These components show how far a vector extends in each direction.
For the vector \(\mathbf{u} = \langle a, b \rangle\):

• \(a\) is the horizontal component, showing movement along the x-axis.
• \(b\) is the vertical component, showing movement along the y-axis.

Understanding vector components is crucial when performing calculations such as vector addition or scalar multiplication.
In our exercise:

• When calculating \(3\mathbf{u}\), each component of \(\mathbf{u}\) is individually multiplied by 3 resulting in new components: \(\langle -6, 15 \rangle\).
• Similarly, \(6\mathbf{v}\) results in components multiplied by 6 to form \(\langle 24, 18 \rangle\).

By focusing on individual components, complex vectors can be more easily understood and manipulated.