Problem 19 - 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. $$|\mathbf{a}+\mathbf{c}|-|\mathbf{a}|-|\mathbf{c}|$$

Step-by-Step Solution

Answer

The result is \( \sqrt{68} - \sqrt{13} - \sqrt{37} \).

Step 1: Find \( \mathbf{a} + \mathbf{c} \)

First, add the corresponding components of vectors \(\mathbf{a}\) and \(\mathbf{c}\). \(\mathbf{a} = \langle 2, 3 \rangle\) and \(\mathbf{c} = \langle 6, -1 \rangle\). Adding them gives:\[\mathbf{a} + \mathbf{c} = \langle 2 + 6, 3 + (-1) \rangle = \langle 8, 2 \rangle\]

Step 2: Compute \( |\mathbf{a} + \mathbf{c}| \)

Find the magnitude of \( \mathbf{a} + \mathbf{c} = \langle 8, 2 \rangle \). The magnitude is given by the formula:\[|\mathbf{a} + \mathbf{c}| = \sqrt{8^2 + 2^2} = \sqrt{64 + 4} = \sqrt{68}\]

Step 3: Compute \( |\mathbf{a}| \)

Find the magnitude of vector \( \mathbf{a} = \langle 2, 3 \rangle \) using the magnitude formula:\[|\mathbf{a}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}\]

Step 4: Compute \( |\mathbf{c}| \)

Find the magnitude of vector \( \mathbf{c} = \langle 6, -1 \rangle \) using the magnitude formula:\[|\mathbf{c}| = \sqrt{6^2 + (-1)^2} = \sqrt{36 + 1} = \sqrt{37}\]

Step 5: Compute \( |\mathbf{a} + \mathbf{c}| - |\mathbf{a}| - |\mathbf{c}| \)

Calculate the expression using the magnitudes found in the previous steps:\[|\mathbf{a} + \mathbf{c}| - |\mathbf{a}| - |\mathbf{c}| = \sqrt{68} - \sqrt{13} - \sqrt{37}\]

Key Concepts

Magnitude of a Vector

The magnitude of a vector is like finding the length of a line segment in geometry. It gives us an idea of how long the vector is from start to finish. To calculate the magnitude of a vector, you apply the Pythagorean theorem, adjusted for vectors. Let's break it down for better understanding. Given a vector \( \mathbf{v} = \langle x, y \rangle \), the magnitude is calculated using the formula: \[|\mathbf{v}| = \sqrt{x^2 + y^2}\]

• Take each component of the vector, square it, and add them together.
• Find the square root of the sum to obtain the magnitude.

Let's consider the vector \( \mathbf{a} = \langle 2, 3 \rangle \). The magnitude is found as follows: \[|\mathbf{a}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}\] This process is a straightforward application of the Pythagorean theorem, applied to the components of the vector.

Vector Addition

Vector addition is essential for understanding how to combine two or more vectors into a single vector. This operation involves summing up the corresponding components of each vector. Importantly, the addition maintains the nature of the vector. Here's what you need to know:

• Each vector component gets added to the corresponding component of the other vector.
• The result is a new vector.

For example, with our vectors \( \mathbf{a} = \langle 2, 3 \rangle \) and \( \mathbf{c} = \langle 6, -1 \rangle \), their sum \( \mathbf{a} + \mathbf{c} \) is given by: \[\mathbf{a} + \mathbf{c} = \langle 2+6, 3+(-1) \rangle = \langle 8, 2 \rangle\] Think of vector addition as moving a point in the 2D space by the amounts specified in the vectors. This helps determine how different movements (vectors) combine to reach a new position.

Algebra of Vectors

The algebra of vectors incorporates rules and operations that allow us to manipulate vectors mathematically. It's a part of linear algebra, focusing on vector-related operations such as addition, subtraction, and finding magnitudes. Here's what's important to note:

• Vectors are added by combining their respective components.
• Subtraction involves adding a negative vector.
• Scalar multiplication scales a vector by a constant, affecting its magnitude but not its direction unless it's negative.

In our problem, we utilized vector addition and calculation of magnitudes in parts:

• First, vectors \( \mathbf{a} \) and \( \mathbf{c} \) were added.
• Then, we found the magnitudes of the resulting vector, and the individual vectors \( \mathbf{a} \) and \( \mathbf{c} \).
• The expression \(|\mathbf{a} + \mathbf{c}| - |\mathbf{a}| - |\mathbf{c}| \) required us to apply both algebraic and geometric reasoning.

Understanding the algebra of vectors forms the basis for tackling more complex vector operations in mathematics and applied sciences.