Question

Row and column vectors \(\vec{u}\) and \(\vec{v}\) are defined. Find the product \(\vec{u} \vec{v},\) where possible. $$ \vec{u}=\left[\begin{array}{ll} 1 & -4 \end{array}\right] \quad \vec{v}=\left[\begin{array}{c} -2 \\ 5 \end{array}\right] $$

Short answer

The product \( \vec{u} \vec{v} \) is \( -22 \).

Step-by-step solution

Identify Vector Dimensions

First, we need to identify the dimensions of the vectors \( \vec{u} \) and \( \vec{v} \). \( \vec{u} \) is a row vector with dimensions \( 1 \times 2 \) and \( \vec{v} \) is a column vector with dimensions \( 2 \times 1 \).

Verify Matrix Multiplication Possible

Check if the inner dimensions of \( \vec{u} \cdot \vec{v} \) match. The inner dimensions are both 2, which means matrix multiplication is possible and will result in a scalar (\( 1 \times 1 \)).

Compute the Dot Product

To find \( \vec{u} \cdot \vec{v} \), compute the sum of the products of corresponding elements: \( 1 \times (-2) + (-4) \times 5 = -2 - 20 \).

Simplify the Expression

Now, simplify the expression: \( -2 - 20 = -22 \). The result of \( \vec{u} \cdot \vec{v} \) is \( -22 \).

Key concepts

Vectors

Vectors are fundamental in linear algebra and many scientific computations. A vector is a mathematical object that has both magnitude and direction. Imagine it as an arrow pointing in space. It can represent quantities such as velocity or force. Vectors can be classified based on their orientation:

• Row vectors: These are horizontal lists of numbers.
• Column vectors: These are vertical lists of numbers.

So, if you are working with data, directions, or any kind of measurements, vectors are useful tools. They help simplify mathematical and physical problems by providing a straightforward way to work with multi-dimensional data.

Dot Product

The dot product is an important operation in vector mathematics. It combines two vectors into a single scalar value. This operation helps in determining the angle between vectors and is commonly used in physics and engineering for calculating work and energy. To find the dot product:

• Multiply the corresponding elements of the vectors.
• Add these products to get a single number (a scalar).

For example, given vectors \( \vec{u} = [1, -4] \) and \( \vec{v} = \begin{bmatrix} -2 \ 5 \end{bmatrix} \), the dot product would be calculated as: \( 1 \times (-2) + (-4) \times 5 = -22 \). Understanding dot products helps in grasping concepts like orthogonality and projections in vector spaces.

Row Vector

Row vectors are a kind of vector where numbers are arranged horizontally. They are often used to represent a series of data points or coefficients in a linear equation. For example, the vector \( \vec{u} = [1, -4] \) is a row vector with two elements. Here:

• The dimension of the vector is known as \( 1 \times 2 \), indicating 1 row and 2 columns.
• Typically used in equations and spreadsheets where data is laid out horizontally.

Row vectors are pivotal in matrix operations, as they often represent coefficients in systems of equations or can be a convenient way to express a data set in a flat, tabular form.

Column Vector

Column vectors, in contrast to row vectors, list numbers vertically. They are frequently used in contexts such as solving linear equations and transformations. For instance, the vector \( \vec{v} = \begin{bmatrix} -2 \ 5 \end{bmatrix} \) is a column vector with two elements. Key points include:

• Its dimension is \( 2 \times 1 \), meaning 2 rows and 1 column.
• Commonly used in operations requiring vector transformations and dimensons.

In matrix multiplication, column vectors help examine linear transformations and systems of equations. They simplify understanding how matrix operations impact linear mappings.