Problem 27 - Precalculus Mathematics for Calculus

Question

Two vectors \(u\) and \(v\) are given. Find their dot product \(\mathbf{U}^{*} \mathbf{V}\). $$\mathbf{u}=6 \mathbf{i}-4 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{v}=\frac{5}{6} \mathbf{i}+\frac{3}{2} \mathbf{j}-\mathbf{k}$$

Step-by-Step Solution

Answer

The dot product is 1.

Step 1: Understand the Formula

To find the dot product of two vectors \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) and \( \mathbf{v} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), you use the formula \( \mathbf{u} \cdot \mathbf{v} = ax + by + cz \). Identify which components are represented by \(a, b, c\) in vector \(\mathbf{u}\) and \(x, y, z\) in vector \(\mathbf{v}\).

Step 2: Identify Components

For the vector \(\mathbf{u} = 6 \mathbf{i} - 4 \mathbf{j} - 2 \mathbf{k}\), the components are \(a = 6\), \(b = -4\), and \(c = -2\). For the vector \(\mathbf{v} = \frac{5}{6} \mathbf{i} + \frac{3}{2} \mathbf{j} - \mathbf{k}\), the components are \(x = \frac{5}{6}\), \(y = \frac{3}{2}\), and \(z = -1\).

Step 3: Plug Components into the Dot Product Formula

Substitute the values into the dot product formula: \[ \mathbf{u} \cdot \mathbf{v} = (6)(\frac{5}{6}) + (-4)(\frac{3}{2}) + (-2)(-1) \]

Step 4: Simplify Each Term

Calculate each term separately:1. \(6 \times \frac{5}{6} = 5\)2. \(-4 \times \frac{3}{2} = -6\)3. \(-2 \times -1 = 2\)

Step 5: Add the Results

Add the results from each term: \[ 5 + (-6) + 2 = 1 \] This is the dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \).

Key Concepts

Vector Components

Understanding vector components is essential for working with vectors in mathematics, physics, and engineering applications. A vector can be thought of as a mathematical entity that has both a magnitude and a direction. In a three-dimensional space like ours, vectors are commonly expressed using unit vectors, which are designated by the letters \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). These unit vectors point along the axes of a coordinate system: \( \mathbf{i} \) along the x-axis, \( \mathbf{j} \) along the y-axis, and \( \mathbf{k} \) along the z-axis.

For instance, if we have a vector \( \mathbf{u} = 6 \mathbf{i} - 4 \mathbf{j} - 2 \mathbf{k} \), the numbers 6, -4, and -2 are the scalar components of the vector along the x, y, and z axes, respectively. Each component tells us how far in the respective direction the vector extends. By breaking down a vector into its components, we can easily work with them mathematically, such as performing operations like addition, subtraction, and dot products. This component-based approach is incredibly helpful for visualizing and solving vector-related problems in calculus and other areas of mathematics.

Calculus

Calculus is the branch of mathematics focused on change and motion, encompassing elements like differentiation and integration. Although the specific exercise about dot products is not directly a calculus problem, understanding how dot products fit into the calculus framework can be very useful. Vectors and dot products often come into play when dealing with vector-valued functions and concepts like vector fields and gradients.

For example, when dealing with curves in space, calculus applies by considering vectors as functions of a parameter and using differentiation to find tangents. In such cases, the concepts like the dot product are used to measure things such as the angle between two vectors, which is particularly valuable when dealing with velocity and acceleration vectors.

• The dot product results in a scalar and gives insight into how two vectors relate directionally.
• In vector calculus, knowing how vectors interact helps in solving optimization problems, finding tangents to curves, and understanding the flow of vector fields.

Overall, calculus gives us the tools to approach and solve complex problems involving vectors, motion, and changes in dynamic systems.

Mathematics Problem-Solving

Mathematics problem-solving is a structured approach to finding solutions to mathematical questions using logical reasoning and mathematical tools. When working with vector problems, like finding the dot product, breaking down the task into manageable steps is a key strategy.

The outlined solution for the dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) illustrates this well. Firstly, understanding that the dot product formula is \( \mathbf{u} \cdot \mathbf{v} = ax + by + cz \) involves recognizing each vector's individual components. By identifying these components clearly, you make the problem simpler to handle.

After identifying these components, the next step is substitution into the formula, followed by performing the arithmetic operations to simplify each part.

• This involves both multiplication of the corresponding components and addition of the resulting values.
• It's crucial to perform each arithmetic operation carefully to ensure accuracy.

Finally, adding the terms gives the final result, which, in this case, reflects how aligned the vectors are in their respective directions.

By honing problem-solving skills in guided steps, students can gain confidence and proficiency in handling complex mathematics tasks involving vectors and other components of calculus.