Problem 4 - Thomas Calculus

Question

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ \mathbf{u}-\mathbf{v} $$

Step-by-Step Solution

Answer

The component form is \( \langle 5, -7 \rangle \) and magnitude is \( \sqrt{74} \).

Step 1: Identify the Components of Vectors

First, recognize the given vectors \( \mathbf{u} = \langle 3, -2 \rangle \) and \( \mathbf{v} = \langle -2, 5 \rangle \). We need to find the component form of \( \mathbf{u} - \mathbf{v} \).

Step 2: Subtract the Vectors

Subtract the corresponding components of \( \mathbf{v} \) from \( \mathbf{u} \). This means computing \( \langle 3, -2 \rangle - \langle -2, 5 \rangle \), which gives:\[\mathbf{u} - \mathbf{v} = \langle 3 - (-2), -2 - 5 \rangle = \langle 3 + 2, -2 - 5 \rangle = \langle 5, -7 \rangle.\]

Step 3: Write the Component Form

The component form of the vector \( \mathbf{u} - \mathbf{v} \) is \( \langle 5, -7 \rangle \).

Step 4: Calculate the Magnitude of the Vector

To find the magnitude of \( \mathbf{u} - \mathbf{v} \), use the formula for the magnitude of a vector \( \langle a, b \rangle \), which is \( \sqrt{a^2 + b^2} \). Thus:\[\text{Magnitude} = \sqrt{5^2 + (-7)^2} = \sqrt{25 + 49} = \sqrt{74}.\]

Step 5: Present the Magnitude as a Simplified Expression

The magnitude (length) of the vector \( \mathbf{u} - \mathbf{v} \) is \( \sqrt{74} \). Since \( \sqrt{74} \) is already in its simplest form, we leave it as is.

Key Concepts

Component Form of a Vector

Vectors often appear in mathematics and physics and can be expressed in different forms. The component form is one such method, where we express the vector in terms of its horizontal (x-axis) and vertical (y-axis) components. To simplify, let's think of it as the 'coordinates' of the vector indicating how far along each axis the vector reaches.

When subtracting vectors, you subtract their individual components separately. In our example, the vectors are \( \mathbf{u} = \langle 3, -2 \rangle \) and \( \mathbf{v} = \langle -2, 5 \rangle \).

• Subtract the x-components: The result is \( 3 - (-2) = 5 \).
• Subtract the y-components: The result is \( -2 - 5 = -7 \).

After performing this calculation, the component form of the vector\( \mathbf{u} - \mathbf{v} \) is \( \langle 5, -7 \rangle \). This result shows how the vector spans 5 units along the x-axis and -7 units along the y-axis. Understanding the component form helps visualize the direction and length of vectors easily.

Magnitude of a Vector

The magnitude of a vector, sometimes referred to as its length, indicates how long the vector is from its start to its endpoint. Think of this as measuring the length of an arrow from its base to the tip.

To find the magnitude, we use the Pythagorean theorem in the setting of the vector's component form. For a vector \( \langle a, b \rangle \), the formula to calculate the magnitude is:\[ \text{Magnitude} = \sqrt{a^2 + b^2} \]In our case, the vector \( \mathbf{u} - \mathbf{v} \) is \( \langle 5, -7 \rangle \). Let's calculate:

• Square each component: \( 5^2 = 25 \) and \( (-7)^2 = 49 \).
• Add these results: \( 25 + 49 = 74 \).
• Take the square root: \( \sqrt{74} \).

So, the magnitude of the vector \( \mathbf{u} - \mathbf{v} \) is \( \sqrt{74} \), giving a precision on how stretched our vector is across the plane.

Vector Operations

Vector operations include processes like addition, subtraction, and scaling, which are essential for manipulating vectors in different applications.**Addition and Subtraction**:

• Vectors are added or subtracted component-wise. This means you combine their x-components together and their y-components together.
• In our example, subtraction is demonstrated, where \( \mathbf{u} = \langle 3, -2 \rangle \) and \( \mathbf{v} = \langle -2, 5 \rangle \). The result \( \mathbf{u} - \mathbf{v} = \langle 5, -7 \rangle \) indicates the differences in components.

**Multiplication by a Scalar**:

• This operation involves multiplying each component of the vector by a single number (called a scalar). This affects the length of the vector but not its direction, unless multiplying by a negative number, which also reverses its direction.

Understanding vector operations expands your ability to analyze and solve problems involving forces, movements, and other vector-related contexts in both mathematics and physics.