Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

For the two-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems \(9-12\), find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle-0.2,0.8\rangle, \mathbf{v}=\langle-2.1,1.3\rangle $$

Step-by-Step Solution

Answer

\( \mathbf{u}+\mathbf{v} = \langle -2.3, 2.1 \rangle\), \( \mathbf{u}-\mathbf{v} = \langle 1.9, -0.5 \rangle\), \( \|\mathbf{u}\| \approx 0.8246 \), \( \|\mathbf{v}\| \approx 2.4698 \).

Step 1: Calculate the Sum of Vectors

To find the sum of two vectors, add their respective components. Given \(\mathbf{u} = \langle -0.2, 0.8 \rangle\) and \(\mathbf{v} = \langle -2.1, 1.3 \rangle\), the sum \(\mathbf{u} + \mathbf{v}\) is calculated as follows:\[ \mathbf{u} + \mathbf{v} = \langle -0.2 + (-2.1), 0.8 + 1.3 \rangle \]Simplifying, we get:\[ \mathbf{u} + \mathbf{v} = \langle -2.3, 2.1 \rangle \]

Step 2: Calculate the Difference of Vectors

To find the difference between two vectors, subtract the components of the second vector from the components of the first vector. Thus, \(\mathbf{u} - \mathbf{v}\) is:\[ \mathbf{u} - \mathbf{v} = \langle -0.2 - (-2.1), 0.8 - 1.3 \rangle \]Simplifying, we have:\[ \mathbf{u} - \mathbf{v} = \langle 1.9, -0.5 \rangle \]

Step 3: Calculate the Magnitude of Vector \(\mathbf{u}\)

The magnitude of a vector \(\mathbf{u} = \langle x, y \rangle\) is given by the formula:\[ \|\mathbf{u}\| = \sqrt{x^2 + y^2} \]For \(\mathbf{u} = \langle -0.2, 0.8 \rangle\), we calculate:\[ \|\mathbf{u}\| = \sqrt{(-0.2)^2 + (0.8)^2} = \sqrt{0.04 + 0.64} = \sqrt{0.68} \approx 0.8246 \]

Step 4: Calculate the Magnitude of Vector \(\mathbf{v}\)

The magnitude of vector \(\mathbf{v} = \langle -2.1, 1.3 \rangle\) is computed similarly as follows:\[ \|\mathbf{v}\| = \sqrt{(-2.1)^2 + (1.3)^2} = \sqrt{4.41 + 1.69} = \sqrt{6.1} \approx 2.4698 \]

Key Concepts

Vector Addition

Adding vectors is like combining forces. Imagine pushing a box in two different directions. When we perform vector addition, we're adding the efforts to find the direction and strength of the total push.

For two-dimensional vectors like - - to perform vector addition, simply add the corresponding components from each vector. In mathematical terms, if \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), then their sum, \( \mathbf{a} + \mathbf{b} \), is obtained as:

• \( \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle \)

For example, for vectors \( \mathbf{u} = \langle -0.2, 0.8 \rangle \) and \( \mathbf{v} = \langle -2.1, 1.3 \rangle \), the addition gives us:

• \( \mathbf{u} + \mathbf{v} = \langle -2.3, 2.1 \rangle \)

This sum yields a new vector that shows the combined direction and magnitude of the two original vectors.

Vector Subtraction

Subtracting vectors is similar to reversing and then adding. It involves figuring out how one vector changes when moving from the other.

To subtract one vector from another, take each component of the second vector away from the matching component of the first vector. For example, if \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the difference \( \mathbf{a} - \mathbf{b} \) is:

• \( \mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2 \rangle \)

For our vectors \( \mathbf{u} = \langle -0.2, 0.8 \rangle \) and \( \mathbf{v} = \langle -2.1, 1.3 \rangle \), this gives:

• \( \mathbf{u} - \mathbf{v} = \langle 1.9, -0.5 \rangle \)

This process reveals a new vector that shows the direction and magnitude of the change from \( \mathbf{u} \) to \( \mathbf{v} \).

Magnitude of a Vector

To understand how strong or long a vector is, we calculate its magnitude. This is like finding how far a person can move if they combine all their steps into one straight line.

The magnitude of a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) can be found using the formula:

• \( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2} \)

Let's apply this formula to our vectors. For \( \mathbf{u} = \langle -0.2, 0.8 \rangle \):

• \( \|\mathbf{u}\| = \sqrt{(-0.2)^2 + (0.8)^2} = \sqrt{0.68} \approx 0.8246 \)

Similarly, for \( \mathbf{v} = \langle -2.1, 1.3 \rangle \):

• \( \|\mathbf{v}\| = \sqrt{(-2.1)^2 + (1.3)^2} = \sqrt{6.1} \approx 2.4698 \)

The magnitude provides a scalar value that tells us how large or strong the vector is.

Two-Dimensional Vectors

Two-dimensional vectors are like arrows on a flat sheet, pointing in a specific direction with a certain length. They help in describing things like motion in a plane or force in physics.

Each two-dimensional vector is defined by two components: one along the x-axis and the other along the y-axis. The vector \( \mathbf{a} \) is typically written as \( \langle a_1, a_2 \rangle \), where:

• \( a_1 \) is the component along the x-axis (horizontal direction)
• \( a_2 \) is the component along the y-axis (vertical direction)

In vector space, vectors like \( \mathbf{u} = \langle -0.2, 0.8 \rangle \) and \( \mathbf{v} = \langle -2.1, 1.3 \rangle \) show us how quantities are directed and measured in two dimensions.
You can visualize these vectors by imagining lines beginning at the origin and ending at the point described by their coordinates. These tools are powerful for understanding dynamics in two-dimensional spaces.