Problem 11 - 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 \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle 12,12\rangle, \mathbf{v}=\langle-2,2\rangle $$

Step-by-Step Solution

Answer

Sum: \( \langle 10, 14 \rangle \), Difference: \( \langle 14, 10 \rangle \), \( \|\mathbf{u}\| = 12\sqrt{2} \), \( \|\mathbf{v}\| = 2\sqrt{2} \).

Step 1: Calculate the Sum of Vectors

To find \( \mathbf{u} + \mathbf{v} \), we add the corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \). The vector \( \mathbf{u} = \langle 12, 12 \rangle \) and \( \mathbf{v} = \langle -2, 2 \rangle \) will be added as: \( \mathbf{u} + \mathbf{v} = \langle 12 + (-2), 12 + 2 \rangle = \langle 10, 14 \rangle \).

Step 2: Calculate the Difference of Vectors

To find \( \mathbf{u} - \mathbf{v} \), subtract the corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \). This gives \( \mathbf{u} - \mathbf{v} = \langle 12 - (-2), 12 - 2 \rangle = \langle 14, 10 \rangle \).

Step 3: Calculate the Magnitude of u

The magnitude of a vector \( \mathbf{u} = \langle 12 , 12 \rangle \) is given by \( \| \mathbf{u} \| = \sqrt{12^2 + 12^2} = \sqrt{144 + 144} = \sqrt{288} = 12\sqrt{2} \).

Step 4: Calculate the Magnitude of v

For \( \mathbf{v} = \langle -2 , 2 \rangle \), the magnitude is \( \| \mathbf{v} \| = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \).

Key Concepts

Vector Addition

Vector addition involves combining two vectors to form a new vector. This is done by adding the corresponding components of the vectors involved. For two-dimensional vectors, like \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), the sum \( \mathbf{u} + \mathbf{v} \) is calculated as follows:

• Add the first components: \( u_1 + v_1 \).
• Add the second components: \( u_2 + v_2 \).

The resulting vector is \( \langle u_1 + v_1, u_2 + v_2 \rangle \). This operation reflects the process of combining the directions and magnitudes of the original vectors. For instance, if you have vectors \( \mathbf{u} = \langle 12, 12 \rangle \) and \( \mathbf{v} = \langle -2, 2 \rangle \), the addition would result in \( \langle 10, 14 \rangle \). This tells us how the overall direction and length are affected when the two vectors are combined.

Vector Subtraction

Vector subtraction is the process of finding the vector difference between two vectors. This is achieved by subtracting the corresponding components of one vector from another. For two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), the result of \( \mathbf{u} - \mathbf{v} \) would be:

• Subtract the first components: \( u_1 - v_1 \).
• Subtract the second components: \( u_2 - v_2 \).

Thus, the resulting vector is \( \langle u_1 - v_1, u_2 - v_2 \rangle \). This action essentially reverses the direction of the vector being subtracted. Applying this to the vectors \( \mathbf{u} = \langle 12, 12 \rangle \) and \( \mathbf{v} = \langle -2, 2 \rangle \), we obtain \( \langle 14, 10 \rangle \). This represents how the vector \( \mathbf{v} \) affects \( \mathbf{u} \) when it is removed from it.

Vector Magnitude

The magnitude of a vector, sometimes referred to as the vector's length, measures its size. It is always a non-negative number. To find the magnitude of a vector \( \mathbf{u} = \langle u_1, u_2 \rangle \), you use the formula:\[\| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2}\]This formula derives from the Pythagorean theorem. By substituting the components of the vector into this formula, we can calculate how long or large the vector is. For example, for the vector \( \mathbf{u} = \langle 12, 12 \rangle \), the magnitude is calculated as \( \sqrt{12^2 + 12^2} = 12\sqrt{2} \). Likewise, for \( \mathbf{v} = \langle -2, 2 \rangle \), the magnitude is \( \sqrt{(-2)^2 + 2^2} = 2\sqrt{2} \). A vector's magnitude is crucial in many applications to understand not just the direction but also the strength or intensity of the vector.

Two-Dimensional Vectors

Two-dimensional vectors are fundamental objects in mathematics and physics representing quantities that have both magnitude and direction. Each vector is expressed as an ordered pair \( \langle x, y \rangle \), where \( x \) and \( y \) are its components along the x-axis and y-axis respectively. This makes them easy to visualize on a Cartesian plane, where any point corresponds to a vector.
This representation allows for the easy application of various vector operations, such as addition, subtraction, and calculating magnitudes. The concept is used in numerous fields to represent positions, velocities, forces, and more. Understanding and manipulating two-dimensional vectors is essential, making them a cornerstone of vector algebra. With practice, they become intuitive tools for describing and solving many types of problems related to motion and spatial geometry.