Question

Row and column vectors \(\vec{u}\) and \(\vec{v}\) are defined. Find the product \(\vec{u} \vec{v},\) where possible. $$ \begin{array}{l} \vec{u}=\left[\begin{array}{llll} 6 & 2 & -1 & 2 \end{array}\right] \\ \vec{v}=\left[\begin{array}{l} 3 \\ 2 \\ 9 \\ 5 \end{array}\right] \end{array} $$

Short answer

The product \(\vec{u} \vec{v}\) is 23.

Step-by-step solution

Understand the Given Vectors

We have a row vector \(\vec{u} = \left[ 6, 2, -1, 2 \right]\) and a column vector \(\vec{v} = \left[ \begin{array}{c} 3 \ 2 \ 9 \ 5 \end{array} \right]\). The row vector has 4 elements and the column vector has 4 elements, making them conformable for multiplication.

Set Up Matrix Multiplication

For the product \(\vec{u} \vec{v}\) to be defined, \(\vec{u}\) must be a row vector and \(\vec{v}\) must be a column vector of the same length. The product will be a scalar value as it results in a 1x1 matrix or single number.

Perform the Dot Product

Multiply corresponding elements of \(\vec{u}\) and \(\vec{v}\), then sum them up: \(6 \cdot 3 + 2 \cdot 2 + (-1) \cdot 9 + 2 \cdot 5 = 18 + 4 - 9 + 10.\)

Calculate the Sum

Adding the results from the element-wise multiplication: \(18 + 4 - 9 + 10 = 23.\)Thus, the dot product \(\vec{u} \vec{v}\) results in 23.

Key concepts

Row Vector

A row vector is simply a one-dimensional array of numbers or elements, arranged in a single row. It's a matrix with one row and multiple columns. In our example, the vector \( \vec{u} = \left[ 6, 2, -1, 2 \right] \) is a row vector. It consists of four elements: 6, 2, -1, and 2.
Row vectors are represented horizontally, and they are often used in mathematical computations like matrix multiplication and transformations.
One way to visualize a row vector is by imagining it as a horizontal list or lineup of numbers, much like a row of seats.

• Use: Row vectors are used in linear algebra to represent coordinates, matrices, or solutions to equations.
• Notation: Typically denoted with square brackets and commas, or spaces between the numbers, like \( \left[ a, b, c \right] \).

Row vectors can be part of a matrix, or they can operate independently, particularly in multiplication procedures where they interact with column vectors.

Column Vector

A column vector is the complementary counterpart to a row vector. It's a one-dimensional array of numbers arranged in a single column. Basically, it's a matrix with one column and multiple rows. In our exercise, \( \vec{v} = \left[ \begin{array}{c} 3 \ 2 \ 9 \ 5 \end{array} \right] \) represents a column vector with four elements.
Column vectors are represented vertically, and they play a key role in various mathematical operations, especially in matrix multiplication.

• Use: Column vectors are often employed to represent data points, solutions in vector spaces, or transformations.
• Notation: They appear with elements stacked vertically, enclosed in brackets or parentheses.

When a column vector interacts with a row vector through multiplication, they can give rise to significant mathematical outcomes, like a scalar.

Dot Product

The dot product is an operation that takes two equal-length sequences of numbers (usually vectors) and returns a single number, known as a scalar. It's also referred to as the scalar product.
To find the dot product, you multiply corresponding elements from each vector, then sum those products. For example, with vectors \( \vec{u} = \left[ 6, 2, -1, 2 \right] \) and \( \vec{v} = \left[ \begin{array}{c} 3 \ 2 \ 9 \ 5 \end{array} \right] \), you compute: \( 6 \cdot 3 + 2 \cdot 2 + (-1) \cdot 9 + 2 \cdot 5 \).
This results in \( 18 + 4 - 9 + 10 = 23 \). The dot product, therefore, is 23.
The dot product:

• Provides a measure of the vectors' magnitude alignment.
• Results in a scalar, simplifying further mathematical computations.

It's vital in applications like determining angles between vectors and finding projections in physics and engineering.

Scalar

A scalar is a single numerical value, often resulting from mathematical operations involving vectors or matrices. Scalars are elements of real numbers, meaning they lack direction or dimension in contrast to vectors.
In our example, after performing the dot product of vectors \( \vec{u} \) and \( \vec{v} \), we arrived at the scalar value 23.
Scalars simplify calculations and are especially significant in physics, where they represent quantities like temperature or mass, providing magnitude but no directional component.

• Key Feature: Represents size or magnitude, often emerging from dot products or other vector transformations.
• Notation: Typically represented as a standalone numeric value, simple and devoid of vector symbols.

Ultimately, understanding scalars in the context of dot products and matrix operations will help you grasp basic linear algebra concepts and their applications.