Problem 17 - Precalculus

Question

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$|\mathbf{b}+\mathbf{c}|$$

Step-by-Step Solution

Answer

The magnitude of \(\mathbf{b} + \mathbf{c}\) is \(\sqrt{130}\).

Step 1: Calculate Vector Sum

First, we need to add vectors \(\mathbf{b}\) and \(\mathbf{c}\). We do this by adding their corresponding components:\[ \mathbf{b} + \mathbf{c} = \langle 5, 4 \rangle + \langle 6, -1 \rangle = \langle 11, 3 \rangle \].

Step 2: Compute Magnitude of Resultant Vector

Next, calculate the magnitude of the resultant vector \(\langle 11, 3 \rangle\). The magnitude of a vector \(\mathbf{v} = \langle x, y \rangle\) is given by \( |\mathbf{v}| = \sqrt{x^2 + y^2} \). Therefore, the magnitude of \(\langle 11, 3 \rangle\) is:\[ |\mathbf{b} + \mathbf{c}| = \sqrt{11^2 + 3^2} = \sqrt{121 + 9} = \sqrt{130} \].

Step 3: Simplify the Magnitude

Finally, simplify \( \sqrt{130} \) if possible. Since 130 cannot be broken down into a product of smaller perfect squares, it is simplified as \( \sqrt{130} \).

Key Concepts

Understanding Vector Addition

Vector addition is an essential operation in vector mathematics. To add two vectors, you combine them based on their corresponding components. Imagine you have two vectors:

• \( \mathbf{b} = \langle 5, 4 \rangle \)
• \( \mathbf{c} = \langle 6, -1 \rangle \)

To find their sum, simply add their respective components: the x-components are added together, as are the y-components. This results in the new vector:\[ \mathbf{b} + \mathbf{c} = \langle 5 + 6, 4 + (-1) \rangle = \langle 11, 3 \rangle \]
This process is a straightforward and practical way to visualize how vectors "combine" their magnitudes and directions.

Calculating the Magnitude of Vectors

The magnitude of a vector is like finding the length of a line. It shows the overall size or extent of the vector in space. After performing vector addition and obtaining a resultant vector, such as \( \langle 11, 3 \rangle \), calculating its magnitude is the next step.
To calculate the magnitude:

• Take the square of each component.
• Add these squares together.
• Take the square root of the result.

So, for vector \( \langle 11, 3 \rangle \), the magnitude is computed as:\[ |\mathbf{r}| = \sqrt{11^2 + 3^2} = \sqrt{121 + 9} = \sqrt{130} \]
This magnitude tells us how long the resultant vector is, straight from the origin to the point (11, 3) in this case.

Exploring Component-Wise Addition

Component-wise addition is a straightforward vector operation. Each vector has components, often noted as \(x\) and \(y\) for two-dimensional vectors. This method simplifies calculations and allows you to work with each part separately.
For example, when adding \( \mathbf{b} = \langle 5, 4 \rangle \) and \( \mathbf{c} = \langle 6, -1 \rangle \), you address:

• The x-components: 5 from \( \mathbf{b} \) and 6 from \( \mathbf{c} \).
• The y-components: 4 from \( \mathbf{b} \) and -1 from \( \mathbf{c} \).

By adding components, you're effectively breaking the problem down into smaller, more manageable pieces.
Thus, resulting in:\[ \mathbf{b} + \mathbf{c} = \langle 5 + 6, 4 + (-1) \rangle = \langle 11, 3 \rangle \]
It's about focusing on each direction separately and understanding how each part contributes to the outcome.