Problem 25 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The dot product is -4.

Step 1: Understand the Dot Product

The dot product of two vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\) is calculated using the formula \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\). This involves multiplying corresponding components and then adding the products.

Step 2: Multiply Corresponding Components

For vectors \(\mathbf{u} = \langle 2,5,0 \rangle\) and \(\mathbf{v} = \left\langle \frac{1}{2},-1,10 \right\rangle\), compute each product: \(2 \times \frac{1}{2}, 5 \times (-1), 0 \times 10\).

Step 3: Calculate Each Product

Compute the individual products: \(2 \times \frac{1}{2} = 1\), \(5 \times (-1) = -5\), and \(0 \times 10 = 0\).

Step 4: Add the Products

Sum the results from Step 3: \(1 + (-5) + 0 = -4\).

Step 5: Provide the Final Result

The dot product \(\mathbf{u} \cdot \mathbf{v}\) is \(-4\).

Key Concepts

Understanding Vectors

Vectors are essential components in mathematics and physics, helping us understand quantities that have both magnitude and direction. Think of vectors as arrows in space, where the length signifies the magnitude, and the direction is indicated by the arrowhead. These are used in various fields such as physics, engineering, and computer graphics to represent forces, velocities, and other directional quantities. A vector is often denoted as \[\mathbf{v} = \langle x, y, z \rangle\]where \(x, y,\) and \(z\) represent the vector's components in three-dimensional space. It's important to recognize:

• Vectors can be added together.
• They can be multiplied by scalars (numbers).
• They allow us to describe real-world phenomena with precision.
• In two dimensions they are often represented as \(\mathbf{v} = \langle x, y \rangle\).

Understanding vectors lays the foundation for more complex operations such as vector multiplication, the focus of which includes dot products.

Exploring Vector Multiplication

Vector multiplication comes in various forms, and the dot product is one of the most fundamental. It combines two vectors to produce a scalar (a single number). This operation, especially handy in determining angles between vectors or projections, simplifies many computational tasks.To multiply two vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\), use the dot product formula:\[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\] Here's the process broken down:

• Multiply each pair of corresponding components from the two vectors.
• Add all these products together to achieve the final scalar value.

For example, if we take \(\mathbf{u} = \langle 2, 5, 0 \rangle\) and \(\mathbf{v} = \left\langle \frac{1}{2}, -1, 10 \right\rangle\), the dot product is calculated as:

• \(2 \times \frac{1}{2} = 1\)
• \(5 \times (-1) = -5\)
• \(0 \times 10 = 0\)

Adding these results gives \(1 + (-5) + 0 = -4\). This result is crucial in various analyses within physics and engineering.

Coordinate Geometry and Its Connection to Vectors

Coordinate geometry, also known as analytic geometry, enables the study of geometry using the coordinate system. By leveraging grids of numbers (coordinates), we can solve geometric problems in a simple, algebraic format. Vectors perfectly align with coordinate geometry as they provide a means to depict geometric elements, like direction and distance on coordinate planes. In coordinate geometry, vectors are used to express:

• Lines and paths by using their direction vectors.
• Position vectors that pinpoint locations in space relative to the origin.
• Relations in non-perpendicular planes, useful in complex geometric problems.

Understanding the relationship between vectors and coordinate geometry allows us to delve deeper into geometry's analytical side, helping solve equations of lines, curves, and even planes, simplifying what might traditionally have been complex problems.