Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle 0,0\rangle, \mathbf{v}=\langle-3,4\rangle $$

Step-by-Step Solution

Answer

The sum is \( \langle -3, 4 \rangle \), the difference is \( \langle 3, -4 \rangle \), \( \|\mathbf{u}\| = 0 \), \( \|\mathbf{v}\| = 5 \).

Step 1: Find the Sum \( \mathbf{u} + \mathbf{v} \)

To find the sum of two vectors, add their respective components. The vector \( \mathbf{u} = \langle 0,0 \rangle \) and the vector \( \mathbf{v} = \langle -3,4 \rangle \). Thus, \( \mathbf{u} + \mathbf{v} = \langle 0+(-3), 0+4 \rangle = \langle -3, 4 \rangle \).

Step 2: Find the Difference \( \mathbf{u} - \mathbf{v} \)

To find the difference between two vectors, subtract their respective components. We again use \( \mathbf{u} = \langle 0,0 \rangle \) and \( \mathbf{v} = \langle -3,4 \rangle \). Thus, \( \mathbf{u} - \mathbf{v} = \langle 0 - (-3), 0 - 4 \rangle = \langle 3, -4 \rangle \).

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

Magnitude of a vector \( \langle a, b \rangle \) is calculated as \( \sqrt{a^2 + b^2} \). For \( \mathbf{u} = \langle 0,0 \rangle \), the magnitude is \( \|\mathbf{u}\| = \sqrt{0^2 + 0^2} = 0 \).

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

Similarly, calculate the magnitude of \( \mathbf{v} = \langle -3,4 \rangle \) using the formula \( \sqrt{a^2 + b^2} \). The magnitude is \( \|\mathbf{v}\| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

Key Concepts

Vector Addition

Vector addition is the process of combining two vectors to create a new vector. You simply add the corresponding components of the vectors. If you have two vectors, say \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), their sum is:

\( \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle \)

This concept is useful in physics and engineering where forces, velocities, and other vector quantities need to be combined. Here's an example calculation:

Given \( \mathbf{u} = \langle 0,0 \rangle \) and \( \mathbf{v} = \langle -3,4 \rangle \), the sum \( \mathbf{u} + \mathbf{v} \) becomes \( \langle 0 + (-3), 0 + 4 \rangle = \langle -3, 4 \rangle \).

This resulting vector \( \langle -3, 4 \rangle \) represents a direction and magnitude combined from the original vectors.

Vector Subtraction

Vector subtraction is similar to vector addition but involves taking away one vector from another. For vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), their difference is:

\( \mathbf{u} - \mathbf{v} = \langle u_1 - v_1, u_2 - v_2 \rangle \)

This operation is useful for determining relative positions or changes. Consider our exercise:

For \( \mathbf{u} = \langle 0,0 \rangle \) and \( \mathbf{v} = \langle -3,4 \rangle \), the difference, \( \mathbf{u} - \mathbf{v} \), results in \( \langle 0 - (-3), 0 - 4 \rangle = \langle 3, -4 \rangle \).

This new vector \( \langle 3, -4 \rangle \) can represent a change in position or direction, highlighting how the result differs from the original vectors.

Vector Magnitude

The magnitude of a vector is one of its key characteristics and reflects the vector's length. Mathematically, for a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \), its magnitude \( \|\mathbf{a}\| \) is calculated using the formula:

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

This measure is important in calculating the strength and impact in various fields such as physics. Let's calculate the magnitudes of the exercise vectors:

For \( \mathbf{u} = \langle 0,0 \rangle \), the magnitude is \( \|\mathbf{u}\| = \sqrt{0^2 + 0^2} = 0 \).

Now the vector \( \mathbf{v} = \langle -3,4 \rangle \):

The magnitude is \( \|\mathbf{v}\| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

These magnitudes tell us that \( \mathbf{v} \) has a substantial length of 5 units, while \( \mathbf{u} \) is essentially a point.

Vectors in Mathematics

Vectors are essential tools in mathematics, helpful in expressing quantities with both direction and magnitude. Unlike simple numbers, vectors offer complex data, commonly displayed as ordered pairs or triplets. **Essentials of Understanding Vectors:**

• Vectors are denoted in component form: \( \langle a, b \rangle \).
• They are visualized as arrows with direction and length, where length corresponds to magnitude.
• Vectors are widely used in physics, engineering, and graphics.

Vectors allow new perspectives on mathematical and physical problems. Understanding vector operations like addition, subtraction, and determining magnitude is key to tackling challenges that involve movement, forces, and various spatial computations.