Question

Find the dot product of the given vectors. \(\vec{u}=\langle 5,3\rangle, \vec{v}=\langle 6,1\rangle\)

Short answer

The dot product is 33.

Step-by-step solution

Understanding the Dot Product Formula

The dot product of two vectors \(\vec{u} = \langle u_1, u_2 \rangle\) and \(\vec{v} = \langle v_1, v_2 \rangle\) is calculated using the formula:\[\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2\]

Identify Components of Vectors

Identify the components of vectors \(\vec{u}\) and \(\vec{v}\). Here, \(\vec{u} = \langle 5, 3 \rangle\) and \(\vec{v} = \langle 6, 1 \rangle\). Thus, \(u_1 = 5\), \(u_2 = 3\), \(v_1 = 6\), and \(v_2 = 1\).

Apply Components to the Dot Product Formula

Substitute the components into the dot product formula:\[\vec{u} \cdot \vec{v} = 5 \times 6 + 3 \times 1\]

Calculate the Products

Calculate each multiplication separately: \(5 \times 6 = 30\) and \(3 \times 1 = 3\).

Sum the Products

Add the results of the products from Step 4:\[30 + 3 = 33\]

Conclude with the Dot Product

The dot product \(\vec{u} \cdot \vec{v}\) is therefore \(33\).

Key concepts

Vector Mathematics

Vector mathematics is a fascinating and fundamental area within the broader field of mathematics and physical sciences. Vectors are quantities that have both magnitude and direction. They can represent anything from a force in physics to position changes in navigation.

In vector mathematics, vectors are typically represented as arrows in a coordinate plane, characterized by components in one or more dimensions. They are often symbolized using bold letters like **u** or with an arrow over the letter, as in \( \vec{u} \). Along with other operations, vector mathematics involves scaling vectors, adding and subtracting them, and one particularly interesting operation called the dot product.

The dot product (also known as scalar product) is an operation that takes two vectors and returns a single scalar value. This is especially useful in projections, determining angles between vectors, and understanding orthogonality. By learning how to find the dot product of vectors, you open up a world of understanding related to the spatial relationships between different directions or forces.

Dot Product Formula

The dot product formula is a straightforward but essential formula in the study of vectors. It allows us to multiply two vectors to find a scalar. Given two vectors \( \vec{u} = \langle u_1, u_2 \rangle \) and \( \vec{v} = \langle v_1, v_2 \rangle \), the dot product is calculated as follows:

\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 \]

This formula is quite intuitive when you understand what it represents. Each component of the first vector is multiplied by the corresponding component of the second vector and then added together. The result is a single number, reflecting how much one vector extends in the direction of the second.

For example, with vectors \( \vec{u} = \langle 5, 3 \rangle \) and \( \vec{v} = \langle 6, 1 \rangle \), applying the dot product formula means multiplying 5 by 6 and adding it to the product of 3 and 1, giving us:

• \( 5 \times 6 = 30 \)
• \( 3 \times 1 = 3 \)

The total is 33, making \( \vec{u} \cdot \vec{v} = 33 \).

This process simplifies the relationship between the two vectors, making it a powerful tool for various applications in mathematics and physics.

Vector Components

Understanding vector components is crucial for mastering vector mathematics and operations like the dot product. In the plane, each vector has different components representing its projections along coordinate axes. For example, a vector in two dimensions, such as \( \vec{u} = \langle 5, 3 \rangle \), consists of two components:

• \( u_1 = 5 \), which is the x-component
• \( u_2 = 3 \), which is the y-component

These components describe how far the vector extends in each direction of the plane.

Similarly, another vector, \( \vec{v} = \langle 6, 1 \rangle \), has its own components: \( v_1 = 6 \) and \( v_2 = 1 \). By examining these components, we see how the vector moves along the x and y axes respectively.

Understanding components is key to performing vector operations. In applying the dot product, for instance, each component from one vector is multiplied by the corresponding component from another vector. Being comfortable with vector components allows you to easily perform operations that are foundational in vector mathematics.