Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Find the dot product of each pair of vectors. $$\langle 2,-3\rangle,\langle 6,5\rangle$$

Step-by-Step Solution

Answer

The dot product is -3.

Step 1: Identify the vectors

We are given two vectors: \( \langle 2, -3 \rangle \) and \( \langle 6, 5 \rangle \). The first vector is represented as \( \mathbf{a} = \langle 2, -3 \rangle \) and the second vector as \( \mathbf{b} = \langle 6, 5 \rangle \).

Step 2: Apply the dot product formula

The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is calculated by the formula: \( \mathbf{a} \cdot \mathbf{b} = a_1 \times b_1 + a_2 \times b_2 \). For these vectors, let \( a_1 = 2 \), \( a_2 = -3 \), \( b_1 = 6 \), and \( b_2 = 5 \).

Step 3: Calculate the product of corresponding components

First, compute the product of the first components: \( a_1 \times b_1 = 2 \times 6 = 12 \). Then, compute the product of the second components: \( a_2 \times b_2 = -3 \times 5 = -15 \).

Step 4: Sum these products

Add the products from Step 3 to get the dot product: \( 12 + (-15) = 12 - 15 = -3 \).

Key Concepts

Vectors

Vectors are fundamental objects in mathematics that hold both magnitude and direction. They are commonly represented in a coordinate system, such as the two-dimensional plane, where a vector is displayed as an arrow pointing from an initial point to a terminal point. In this representation, a vector is expressed in component form, like \( \langle a_1, a_2 \rangle \), where \( a_1 \) and \( a_2 \) are the horizontal and vertical components, respectively. For example, consider a vector \( \langle 2, -3 \rangle \):

• \( a_1 = 2 \) represents the movement along the x-axis.
• \( a_2 = -3 \) represents movement along the y-axis, emphasizing direction with a negative sign.

Vectors are foundational in fields like physics and engineering, where they describe forces, velocities, and other quantities.

Vector Multiplication

Vector multiplication can occur in different ways, but one of the most common is the dot product. This type of multiplication results in a scalar, or single number, hence the term scalar product is also used. The dot product is vital for calculating projections and understanding the relationship between vectors, especially in determining angles and verifying orthogonality. In our exercise with vectors \( \langle 2, -3 \rangle \) and \( \langle 6, 5 \rangle \):

• We multiply the corresponding components for each vector.
• The operation involves a simple multiplication following the order of the given components.

This method allows us to move from the geometric intuition of vectors to a more numerical and analytical perspective.

Dot Product Formula

The dot product formula is a staple in vector mathematics. This computation was shown in our exercise as: \( \mathbf{a} \cdot \mathbf{b} = a_1 \times b_1 + a_2 \times b_2 \). It's a sum of products from paired vector components. In simpler terms:

• Multiply the first components of each vector.
• Multiply the second components.
• Add these products to obtain the dot product.

For the vectors given, this results in \( 2 \times 6 + (-3) \times 5 = 12 - 15 = -3 \). The dot product gives insight into the directional alignment of the vectors. If the dot product is zero, vectors are perpendicular.

Mathematical Calculations

Mathematical calculations are procedures or operations that involve numbers and symbols to determine quantities. They are critical in understanding concepts like the dot product. In our context, calculations involve:

• Accurate multiplication of components from each vector.
• Adding those results to find a single scalar value.
• Attention to signs and arithmetic rules, as seen in \( 12 - 15 = -3 \).

These operations highlight the precision necessary in mathematics to achieve correct and meaningful results. Mastering calculations in dot products helps form a solid base for more advanced topics in vector algebra and linear transformations.