Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Find the dot product of each pair of vectors. $$2 \mathbf{i}+4 \mathbf{j},-\mathbf{j}$$

Step-by-Step Solution

Answer

The dot product is -4.

Step 1: Understand the Dot Product

The dot product (also known as scalar product) is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. For vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the dot product is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

Step 2: Identify Components of the Vectors

For the first vector \( 2 \mathbf{i} + 4 \mathbf{j} \), the components are \( a_1 = 2 \) and \( a_2 = 4 \). For the second vector \( -\mathbf{j} \), the components are \( b_1 = 0 \) and \( b_2 = -1 \).

Step 3: Apply the Dot Product Formula

Use the formula \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \). Insert the known components: \( 2 \times 0 + 4 \times (-1) \).

Step 4: Calculate the Dot Product

First, compute \( 2 \times 0 = 0 \). Next, compute \( 4 \times (-1) = -4 \). Combine the results: \( 0 + (-4) = -4 \).

Key Concepts

Vectors

Vectors are fundamental elements in mathematics and physics used to represent quantities that have both a magnitude and a direction. Think of vectors as arrows pointing from one point to another in space. Each vector is defined by two main properties:

• Magnitude: The length or size of the vector.
• Direction: The orientation of the vector relative to a chosen reference system.

When dealing with vectors in a two-dimensional plane, we often break them down into components based on the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\), which point in the horizontal and vertical directions, respectively.

For example, a vector \( \mathbf{a} = 2\mathbf{i} + 4\mathbf{j} \) means it goes 2 units along the x-axis and 4 units along the y-axis. This visual and numerical representation helps simplify many calculations, especially when performing operations like the dot product.

Coordinate Vectors

Coordinate vectors are a simplified representation of vectors using numerical components for easy computation. In a 2D space, a vector can be represented as \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \), where \(a_1\) and \(a_2\) represent the vector's horizontal and vertical components, respectively.

These components essentially tell us how far to move along each axis from the origin.
For example:

• The vector \(2\mathbf{i} + 4\mathbf{j}\) can be broken down into \((2, 4)\). This indicates moving 2 units in the x-direction and 4 units in the y-direction.
• The vector \(-\mathbf{j}\) can be expressed as \((0, -1)\). This suggests no movement along the x-axis and a downward movement of 1 unit along the y-axis.

Coordinate vectors provide a convenient way to handle vector mathematics, as we commonly deal with numbers instead of symbolic vector components.

Scalar Product

The scalar product, more commonly known as the dot product, is a method used to multiply two vectors, particularly in physics and geometry, to find how much one vector extends in the direction of another. Unlike other multiplication methods that yield a resultant vector, the dot product results in a scalar—a single numerical value.

The dot product is defined for two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) by the formula:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]
This calculation captures the idea that only the components of the vectors in the same direction contribute to the dot product.

In our exercise, the vectors are \(2 \mathbf{i} + 4 \mathbf{j}\) and \(-\mathbf{j}\). Implementing the dot product formula:

• \(a_1 = 2\), \(a_2 = 4\)
• \(b_1 = 0\), \(b_2 = -1\)
• The operation \(2 \times 0 + 4 \times (-1)\) results in \(-4\).

This result means that, in a coordinate sense, two vectors' interaction in their overlapping directions adds up to \(-4\). The beauty of the dot product lies in its ability to convey such directional relationships succinctly.