Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Find the dot product of each pair of vectors. $$\langle 6,-1\rangle,\langle 2,5\rangle$$

Step-by-Step Solution

Answer

The dot product is 7.

Step 1: Identify the Components of Each Vector

To solve this problem, we first need to identify the components of both vectors. The first vector is \( \langle 6, -1 \rangle \), and the second vector is \( \langle 2, 5 \rangle \).

Step 2: Apply the Dot Product Formula

The formula for the dot product of two vectors \( \langle a_1, b_1 \rangle \) and \( \langle a_2, b_2 \rangle \) is given by: \( a_1 \cdot a_2 + b_1 \cdot b_2 \).

Step 3: Multiply Corresponding Components

Substitute the components from the vectors into the formula: \( (6 \cdot 2) + (-1 \cdot 5) \). Calculate the products to get \( 12 + (-5) \).

Step 4: Sum the Products

Finally, add the products together: \( 12 - 5 = 7 \). This value is the dot product of the two vectors.

Key Concepts

Vectors

Vectors are fundamental elements in mathematics and physics, describing both magnitude and direction. They can be visualized as arrows pointing from one location to another. In a two-dimensional space, vectors have two elements called components, typically represented in the form \( \langle a, b \rangle \). These components describe how far the vector goes in each direction: \( a \) units in the x-direction and \( b \) units in the y-direction.

For example, if you think of driving in a car, a vector can describe how far north and east you travel. It not only tells you how far you have gone, but also in which direction. Vectors are not limited to just two dimensions; however, in this exercise, we only work with two-dimensional vectors.

Components of Vectors

Understanding vector components is essential for calculations involving vectors, such as finding the dot product. Each vector component helps to precisely define its direction and magnitude on a coordinate plane.

Consider a vector \( \langle 6, -1 \rangle \). This representation tells us that the vector moves 6 units along the positive x-axis and 1 unit along the negative y-axis. Similarly, another vector \( \langle 2, 5 \rangle \) moves 2 units along the positive x-axis and 5 units along the positive y-axis.

• The first number in the pair is the x-component, detailing movement along the x-axis.
• The second number in the pair is the y-component, detailing movement along the y-axis.

By knowing these components, it becomes straightforward to perform vector calculations.

Dot Product Formula

The dot product is a valuable operation that results in a single number from two vectors. The formula is straightforward: for vectors \( \langle a_1, b_1 \rangle \) and \( \langle a_2, b_2 \rangle \), the dot product is computed as \( a_1 \cdot a_2 + b_1 \cdot b_2 \).

To apply this formula to our vectors \( \langle 6, -1 \rangle \) and \( \langle 2, 5 \rangle \):

• First, multiply their corresponding components: \((6 \times 2) + (-1 \times 5)\).
• This results in \(12 + (-5)\).
• Finally, sum these products: \(12 - 5 = 7\).

In this context, the number 7 represents the dot product of these two vectors. It's a measure of how much one vector extends in the direction of another.