Problem 16 - 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 \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle 0.3,0.3,0.5\rangle, \mathbf{v}=\langle 2.2,1.3,-0.9\rangle $$

Step-by-Step Solution

Answer

\(\mathbf{u}+\mathbf{v} = \langle 2.5, 1.6, -0.4 \rangle\), \(\mathbf{u}-\mathbf{v} = \langle -1.9, -1.0, 1.4 \rangle\), \(\|\mathbf{u}\| \approx 0.656\), \(\|\mathbf{v}\| \approx 2.71\).

Step 1: Adding the Vectors

To find the sum \( \mathbf{u} + \mathbf{v} \), add the corresponding components of the vectors \( \mathbf{u} = \langle 0.3, 0.3, 0.5 \rangle \) and \( \mathbf{v} = \langle 2.2, 1.3, -0.9 \rangle \).\[\mathbf{u} + \mathbf{v} = \langle 0.3 + 2.2, 0.3 + 1.3, 0.5 + (-0.9) \rangle = \langle 2.5, 1.6, -0.4 \rangle\]The sum of the vectors is \( \langle 2.5, 1.6, -0.4 \rangle \).

Step 2: Subtracting the Vectors

To find the difference \( \mathbf{u} - \mathbf{v} \), subtract the corresponding components of vector \( \mathbf{v} \) from vector \( \mathbf{u} \).\[\mathbf{u} - \mathbf{v} = \langle 0.3 - 2.2, 0.3 - 1.3, 0.5 - (-0.9) \rangle = \langle -1.9, -1.0, 1.4 \rangle\]The difference of the vectors is \( \langle -1.9, -1.0, 1.4 \rangle \).

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

To find the magnitude of vector \( \mathbf{u} \), use the formula \( \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2} \).\[\|\mathbf{u}\| = \sqrt{(0.3)^2 + (0.3)^2 + (0.5)^2} = \sqrt{0.09 + 0.09 + 0.25} = \sqrt{0.43}\]The magnitude of \( \mathbf{u} \) is approximately \( 0.656 \).

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

To find the magnitude of vector \( \mathbf{v} \), use the formula \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \).\[\|\mathbf{v}\| = \sqrt{(2.2)^2 + (1.3)^2 + (-0.9)^2} = \sqrt{4.84 + 1.69 + 0.81} = \sqrt{7.34}\]The magnitude of \( \mathbf{v} \) is approximately \( 2.71 \).

Key Concepts

Vector Addition

Vector addition is a fundamental concept in mathematics and physics, where we combine two vectors to get a resultant vector. To add two vectors, we simply add their corresponding components.
For example, with vectors \( \mathbf{u} = \langle 0.3, 0.3, 0.5 \rangle \) and \( \mathbf{v} = \langle 2.2, 1.3, -0.9 \rangle \):

• Add the x-components: \( 0.3 + 2.2 = 2.5 \)
• Add the y-components: \( 0.3 + 1.3 = 1.6 \)
• Add the z-components: \( 0.5 + (-0.9) = -0.4 \)

The resulting vector \( \mathbf{u} + \mathbf{v} \) is \( \langle 2.5, 1.6, -0.4 \rangle \). This vector represents moving by the combined distance and direction of both original vectors. The simplicity of vector addition makes it a useful tool in physics and engineering for calculating combined movements or forces.

Vector Subtraction

Vector subtraction is a little like reversing addition. It allows us to find the difference between two vectors, which can represent relative motion or separation between two points in space.
To subtract vector \( \mathbf{v} \) from vector \( \mathbf{u} \) follows a similar process as addition, only we subtract each component one by one:

• Subtract the x-components: \( 0.3 - 2.2 = -1.9 \)
• Subtract the y-components: \( 0.3 - 1.3 = -1.0 \)
• Subtract the z-components: \(0.5 - (-0.9) = 1.4 \)

Hence, the result of \( \mathbf{u} - \mathbf{v} \) is \( \langle -1.9, -1.0, 1.4 \rangle \). This operation is especially useful in scenarios where we need to compute the result of a displacement from one vector to another or analyze relative velocities.

Magnitude of a Vector

The magnitude of a vector is a measure of its length. In three-dimensional space, the magnitude is found using the Pythagorean theorem extended into three dimensions.
The magnitude, \( \|\mathbf{u}\| \) of vector \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) is calculated as:

• \( \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2} \)

For our vector \( \mathbf{u} = \langle 0.3, 0.3, 0.5 \rangle \), the magnitude is:\[ \|\mathbf{u}\| = \sqrt{(0.3)^2 + (0.3)^2 + (0.5)^2} = \sqrt{0.43}, \approx 0.656 \]
Similarly, for vector \( \mathbf{v} \):\[ \|\mathbf{v}\| = \sqrt{(2.2)^2 + (1.3)^2 + (-0.9)^2} = \sqrt{7.34}, \approx 2.71 \]
The magnitude gives us the length from the origin in a coordinate system to the point defined by the vector components, often used to describe physical quantities like force or velocity.

Components of Vectors

Vectors consist of components, which determine their behavior in a given coordinate space. Each component contributes uniquely to the direction and magnitude of a vector. When we describe a vector in terms of components, we use coordinates, typically in the form \( \langle x, y, z \rangle \).
For example, vector \( \mathbf{u} = \langle 0.3, 0.3, 0.5 \rangle \) is broken down into:

• An x-component of 0.3
• A y-component of 0.3
• A z-component of 0.5

Each component relates to movement or extension along its respective axis. This breakdown makes it easier to apply mathematical operations on vectors, such as addition, subtraction, and finding magnitude. Components play a vital role in understanding and solving many real-world problems where direction and magnitude are crucial, like in navigation and physics calculations.