Problem 25 - Precalculus Mathematics for Calculus

Question

Two vectors \(u\) and \(v\) are given. Find their dot product \(\mathbf{U}^{*} \mathbf{V}\). $$\mathbf{u}=\langle 2,5,0\rangle, \quad \mathbf{v}=\left\langle\frac{1}{2},-1,10\right\rangle$$

Step-by-Step Solution

Answer

The dot product is \(-4\).

Step 1: Understanding Dot Product

The dot product of two vectors is found by multiplying corresponding components of the vectors and then summing up the products. If vectors \( \mathbf{u} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{v} = \langle b_1, b_2, b_3 \rangle \), then the dot product is \( a_1b_1 + a_2b_2 + a_3b_3 \).

Step 2: Component-wise Multiplication

For vectors \( \mathbf{u} = \langle 2, 5, 0 \rangle \) and \( \mathbf{v} = \langle \frac{1}{2}, -1, 10 \rangle \), compute the component-wise products: 1. \(2 \times \frac{1}{2} = 1\)2. \(5 \times (-1) = -5\)3. \(0 \times 10 = 0\).

Step 3: Summing the Products

Now add the results of each component multiplication together: \[1 + (-5) + 0 = 1 - 5 = -4\]. Therefore, the dot product of \( \mathbf{u} \) and \( \mathbf{v} \) is \(-4\).

Key Concepts

vector multiplication

In mathematics, multiplying vectors is not as straightforward as regular number multiplication. One of the most common forms of vector multiplication is the dot product. This operation focuses on combining two vectors to produce a single scalar value. The dot product is not only a fundamental concept in vector mathematics but also widely used in various fields like physics and engineering to compute things like force, energy, and work. To calculate the dot product of two vectors, say \( \mathbf{u} \) and \( \mathbf{v} \), each component of \( \mathbf{u} \) is multiplied by its corresponding component in \( \mathbf{v} \).
Once each pair of components is multiplied, these products are summed together. This sum gives you the dot product, a real, single number.

When performing vector multiplication through the dot product:

• Ensure vectors are of equal dimensions to use the dot product.
• The result is a scalar, not a vector, making its interpretation straightforward in calculations.
• Useful in determining vector angles, orthogonality, and projections between vectors.

component-wise product

The component-wise product lays the groundwork for the dot product calculation. This step is simple as each component of the first vector is individually multiplied by the corresponding component of the second vector. For example, if you have two vectors, \( \mathbf{u} = \langle 2, 5, 0 \rangle \) and \( \mathbf{v} = \langle \frac{1}{2}, -1, 10 \rangle \), you multiply component by component:

• First component: \(2 \times \frac{1}{2} = 1\)
• Second component: \(5 \times (-1) = -5\)
• Third component: \(0 \times 10 = 0\)

The component-wise product does not give the final result of vector multiplication directly but is a crucial intermediate step toward it. Each component's multiplication makes it easier to handle and visualize the operation involved, ensuring clarity and reducing the possibility of errors.
Understanding this process helps in managing both theoretical and practical applications, simplifying complex vector mathematics into manageable calculations.

vector operations

Vector operations become quite fundamental when dealing with problems involving physics or engineering. Vectors can describe quantities that not only have a magnitude but also a direction, such as velocity or force. Vector operations allow us to add, subtract, multiply, and analyze these quantities.

Primarily, vector operations include:

• Addition: Summing corresponding components of two vectors.
• Subtraction: Subtracting corresponding components.
• Multiplication: Includes scalar multiplication and the dot product, which we explored earlier.
• Cross Product: Combining vectors to form another vector perpendicular to the original pair (defined only in 3-dimensional space).

Each operation has its specific use and rules that govern how they affect vectors. For instance, vector multiplication through the dot product is used not just for solving equations, but also for determining how parallel two vectors are. When the dot product is zero, vectors are orthogonal, implying a significant geometric interpretation.
By learning these operations and understanding how they manipulate vector quantities, you gain valuable insight into both theoretical and practical problems involving vector mathematics.