Product

1. Scalar Multiplication

Elements Involved:

• A scalar (a single number) and a vector or matrix.

How to Calculate:

• Multiply each element of the vector or matrix by the scalar.

Example (with a vector):

Let $\vec{v}=\left[\begin{array}{c}2\\ 4\end{array}\right]$ and the scalar be $3$:

$$3\times \vec{v}=3\times \left[\begin{array}{c}2\\ 4\end{array}\right]=\left[\begin{array}{c}6\\ 12\end{array}\right]$$

Result:

• The result is a vector or matrix of the same size as the original, but each element is multiplied by the scalar.

---

2. Dot Product (or Inner Product)

Elements Involved:

• Two vectors of the same size.

How to Calculate:

• Multiply the corresponding elements of the vectors together and then sum the results.

Example (with two 3D vectors):

Let $\vec{a}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ and $\vec{b}=\left[\begin{array}{c}4\\ 5\\ 6\end{array}\right]$:

$$\vec{a}\cdot \vec{b}=(1\times 4)+(2\times 5)+(3\times 6)=4+10+18=32$$

Result:

• The result is a scalar (a single number).
• The dot product measures the alignment of two vectors: if the result is 0, the vectors are orthogonal.

---

3. Matrix Multiplication

Elements Involved:

• Two matrices, but their dimensions must satisfy the following condition: if matrix $A$ is of size $m\times n$, matrix $B$ must be of size $n\times p$. In other words, the number of columns of the first matrix must equal the number of rows of the second matrix.

How to Calculate:

• Multiply the elements of each row of the first matrix by the corresponding elements of each column of the second matrix, and then sum the results for each pair. The element in the $i^{th}$ row and $j^{th}$ column of the result matrix is the dot product of the $i^{th}$ row of the first matrix and the $j^{th}$ column of the second matrix.

Example:

Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$:

$$A\times B=\left[\begin{array}{cc}(1\times 5+2\times 7)& (1\times 6+2\times 8)\\ (3\times 5+4\times 7)& (3\times 6+4\times 8)\end{array}\right]=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$$

Result:

• The result is a matrix with dimensions determined by the rows of the first matrix and the columns of the second matrix. In this case, $A$ is $2\times 2$ and $B$ is $2\times 2$, so the result is a $2\times 2$ matrix.

---

4. Cross Product

Elements Involved:

• Two 3-dimensional vectors.

How to Calculate:

• The cross product of two vectors $\vec{a}$ and $\vec{b}$, written as $\vec{a}\times \vec{b}$, is a new vector that is perpendicular to both $\vec{a}$ and $\vec{b}$, following the right-hand rule.
• For $\vec{a}=\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]$ and $\vec{b}=\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$, the cross product is:

$$\vec{a}\times \vec{b}=\left[\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right]$$

Example:

Let $\vec{a}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ and $\vec{b}=\left[\begin{array}{c}4\\ 5\\ 6\end{array}\right]$:

$$\vec{a}\times \vec{b}=\left[\begin{array}{c}(2\times 6-3\times 5)\\ (3\times 4-1\times 6)\\ (1\times 5-2\times 4)\end{array}\right]=\left[\begin{array}{c}-3\\ 6\\ -3\end{array}\right]$$

Result:

• The result is a new 3-dimensional vector that is perpendicular to the original vectors. The direction of the resulting vector is determined by the right-hand rule.

---

Summary:

Operation	Elements Involved	How to Calculate	Result Type
Scalar Multiplication	Scalar and Vector/Matrix	Multiply each element by the scalar	Vector or Matrix
Dot Product	Two Vectors (same size)	Multiply corresponding elements and sum	Scalar (single number)
Matrix Multiplication	Two Matrices (compatible dimensions)	Multiply rows by columns and sum	Matrix
Cross Product	Two 3D Vectors	Use the formula for each component of the new vector	Vector (3D)