Problem 11 - Precalculus

Question

Find the indicated dot product. $$\langle 0.8,-0.5\rangle \cdot\langle 2,6\rangle$$

Step-by-Step Solution

Answer

The dot product is -1.4.

Step 1: Understand the Dot Product

The dot product of two vectors is a scalar obtained by multiplying corresponding entries of the vectors and then summing those products. If you have vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), then the dot product \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

Step 2: Identify Components of Vectors

Identify the components of the two vectors given in the problem. For \( \langle 0.8, -0.5 \rangle \), the components are \( 0.8 \) and \( -0.5 \). For \( \langle 2, 6 \rangle \), the components are \( 2 \) and \( 6 \).

Step 3: Multiply Corresponding Components

Multiply the corresponding components of the vectors together: \( 0.8 \times 2 \) and \( -0.5 \times 6 \).

Step 4: Calculate Products

First calculation: \( 0.8 \times 2 = 1.6 \).Second calculation: \( -0.5 \times 6 = -3 \).

Step 5: Sum the Results

Add the two products obtained from the multiplications: \( 1.6 + (-3) \).

Step 6: Final Calculation

Compute the sum: \( 1.6 - 3 = -1.4 \).

Key Concepts

Vectors

Vectors are mathematical objects characterized by having both a direction and a magnitude. They are utilized in many branches of science, including physics and engineering, to describe quantities like force or velocity. Vectors are often represented in the form of coordinate pairs or triples, like \( \langle x, y \rangle \) in two dimensions.

• Direction: Indicates where the vector is pointing.
• Magnitude: Represents the length or size of the vector.

In our problem, we have the vectors \( \langle 0.8,-0.5\rangle \) and \( \langle 2,6 \rangle \). They help us understand how two quantities can interact in a spatial setting, which is a fundamental concept in physics and engineering. Remember, vectors can be added, scaled, and used to find various products, such as the dot product.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is simply a real number. This operation affects the magnitude of the vector but does not alter its direction. For instance, if a vector is multiplied by a positive scalar, the result is a vector that points in the same direction but differs in scale.

• Doubling a vector: If a vector is multiplied by \( 2 \), all components are doubled, extending its length without altering its path.
• Negating a vector: Multiplying by a negative scalar will reverse the vector's direction.

When calculating the dot product, as seen in our example, scalar multiplication happens when we multiply each pair of corresponding components. For instance, multiplying \( 0.8 \) by \( 2 \) or \( -0.5 \) by \( 6 \). The operation transforms two vectors into scalar quantities, laying the foundation for additional computations.

Vector Components

Vector components are individual elements that make up a vector. Each vector has several components (two in 2D, three in 3D) which dictate its characteristics.
In the context of the original exercise, we deal with two vectors: \( \langle 0.8, -0.5 \rangle \) and \( \langle 2, 6 \rangle \).

• First vector components: \( 0.8 \) (horizontal) and \( -0.5 \) (vertical).
• Second vector components: \( 2 \) (horizontal) and \( 6 \) (vertical).

Understanding these components lets us break down complex vectors into manageable parts, simplifying calculations like the dot product. It assists in decoding how each part contributes to the vector's overall behavior. In the dot product calculation, each pair of components from the respective vectors is crucial for computing the final scalar result.