Problem 46 - Precalculus with Calculus Previews

Question

Find the dot product \(\mathbf{u} \cdot \mathbf{v}\). $$ \mathbf{u}=\langle 1,-2\rangle, \mathbf{v}=\langle 4,0\rangle $$

Step-by-Step Solution

Answer

The dot product is 4.

Step 1: Understand the Dot Product Formula

The dot product (also known as the scalar product) of two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) is calculated as follows: \[ \mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2 \]This means you multiply the corresponding components of the vectors and then add the results.

Step 2: Substitute the Vector Components

Substitute the components of vectors \( \mathbf{u} \) and \( \mathbf{v} \) into the dot product formula:\( u_1 = 1, u_2 = -2, v_1 = 4, v_2 = 0 \).This will result in: \[ \mathbf{u} \cdot \mathbf{v} = (1) \cdot (4) + (-2) \cdot (0) \]

Step 3: Multiply the Components

Perform the multiplications:- Multiply the first components: \( 1 \cdot 4 = 4 \).- Multiply the second components: \( -2 \cdot 0 = 0 \).

Step 4: Add the Products

Add the results of the multiplication together: \[ 4 + 0 = 4 \]

Step 5: Final Answer

The dot product of \( \mathbf{u} \) and \( \mathbf{v} \) is 4.

Key Concepts

Vectors

Understanding vectors is crucial when working with the dot product. A vector is essentially a quantity that has both direction and magnitude.
They can be represented visually as arrows in a coordinate system, where the length of the arrow indicates the magnitude, and the direction in which it points represents the direction.

• Notation: Vectors are often written in angle brackets, like \( \langle a, b \rangle \), where \( a \) and \( b \) are the components in the horizontal and vertical directions respectively.
• Examples: The vectors in our problem are \( \mathbf{u} = \langle 1, -2 \rangle \) and \( \mathbf{v} = \langle 4, 0 \rangle \).

To work with vectors, you should be familiar with these components and how they allow us to calculate other vector operations, such as the dot product.
Remember, the direction and magnitude of the vector tell us different things about its influence in calculations.

Vector Components

Vector components are the building blocks when dealing with vectors. Each vector has components that show how far along each axis the vector reaches.
In a two-dimensional vector like \( \langle u_1, u_2 \rangle \), \( u_1 \) is the horizontal component, and \( u_2 \) is the vertical component.

• Calculating Components: To analyze a vector, it's important to know these components as they define the vector's position in space.
• Problem Example: For the vectors \( \mathbf{u} \) and \( \mathbf{v} \), the components are \( u_1 = 1 \), \( u_2 = -2 \), \( v_1 = 4 \), and \( v_2 = 0 \).

When calculating things like the dot product, these components are multiplied and then summed.
This shows just how essential components are to understand and work correctly with vectors.

Scalar Product

The scalar product, also known as the dot product, is a significant vector operation. This operation converts two vectors into a single number (a scalar), which provides information on their directional alignment.

• Formula: The dot product formula is \( \mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2 \).
• Purpose: The result tells you how closely the two vectors point in the same direction. If the result is zero, the vectors are perpendicular.
• Example Calculation: By substituting components from our problem, \( (1) \cdot (4) + (-2) \cdot (0) = 4 \), we calculated a dot product of 4.

The scalar product is particularly used in physics and engineering to quantify aspects like work done or projecting one vector onto another.
Understanding this concept is key to master mathematics and its applications.