Norm, Unit Vectors, and Gram Determinant in College Algebra

Norm and Angle

Definition of Norm (Length) of a Vector

The norm (or length) of a vector is a measure of its magnitude in space. For a vector in two or more dimensions, the norm is calculated using the Pythagorean theorem.

• For a vector $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ in $\mathbb{R}^2$:

$\|\vec{x}\| = \sqrt{x_1^2 + x_2^2}$

• For a vector $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ in $\mathbb{R}^n$:

$\|\vec{x}\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}$

• Key Point: The norm is always non-negative and equals zero only for the zero vector.

Example: For $\vec{x} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, $\|\vec{x}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Theorem: Norm and Dot Product

The dot product of a vector with itself equals the square of its norm.

$\vec{x} \cdot \vec{x} = \|\vec{x}\|^2$

• Proof Outline: For $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ \cdots \\ x_n \end{bmatrix}$, $\vec{x} \cdot \vec{x} = x_1^2 + x_2^2 + \cdots + x_n^2 = \|\vec{x}\|^2$.

Example: For $\vec{x} = \begin{bmatrix} 2 \\ 1 \\ 4 \\ 2 \end{bmatrix}$, $\vec{x} \cdot \vec{x} = 2^2 + 1^2 + 4^2 + 2^2 = 4 + 1 + 16 + 4 = 25$.

Finding the Length of a Vector Between Two Points

To find the length of a vector defined by two points $P$ and $Q$ in $\mathbb{R}^n$, subtract the coordinates of $P$ from $Q$ to get the vector $\vec{v} = Q - P$, then compute its norm.

• Formula: If $P = (p_1, p_2, ..., p_n)$ and $Q = (q_1, q_2, ..., q_n)$, then $\vec{v} = Q - P = (q_1 - p_1, q_2 - p_2, ..., q_n - p_n)$ and $\|\vec{v}\| = \sqrt{(q_1 - p_1)^2 + (q_2 - p_2)^2 + \cdots + (q_n - p_n)^2}$.

Example: For $P = (-1, 3, 4)$ and $Q = (2, -5, 1)$, $Q - P = (3, -8, -3)$, so $\|Q - P\| = \sqrt{3^2 + (-8)^2 + (-3)^2} = \sqrt{9 + 64 + 9} = \sqrt{82}$.

Unit Vectors (Normalizing a Vector)

Definition and Construction

A unit vector is a vector with length 1. To find a unit vector in the direction of a nonzero vector $\vec{x}$, divide $\vec{x}$ by its norm:

$\hat{x} = \dfrac{\vec{x}}{\|\vec{x}\|}$

• Key Point: The unit vector $\hat{x}$ has the same direction as $\vec{x}$ but a magnitude of 1.

Example: For $\vec{x} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, $\|\vec{x}\| = 5$, so the unit vector is $\hat{x} = \dfrac{1}{5} \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 3/5 \\ 4/5 \end{bmatrix}$.

• Verification: $\|\hat{x}\| = \sqrt{(3/5)^2 + (4/5)^2} = \sqrt{9/25 + 16/25} = \sqrt{25/25} = 1$.

Gram Determinant

Determinant of a 2x2 Matrix

The determinant of a $2 \times 2$ matrix $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ is calculated as:

$\det(A) = a_{11}a_{22} - a_{12}a_{21}$

Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $\det(A) = (1)(4) - (2)(3) = 4 - 6 = -2$.

Definition: Gram Determinant

The Gram determinant of two vectors $\vec{a}$ and $\vec{b}$ in $\mathbb{R}^n$ is defined as:

$G(\vec{a}, \vec{b}) = \|\vec{a}\|^2 \|\vec{b}\|^2 - (\vec{a} \cdot \vec{b})^2$

• Purpose: The Gram determinant measures the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$ in $\mathbb{R}^2$ or the volume in higher dimensions.

Example: If $\vec{a} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$, then $\|\vec{a}\| = 1$, $\|\vec{b}\| = 1$, $\vec{a} \cdot \vec{b} = 0$, so $G(\vec{a}, \vec{b}) = 1^2 \cdot 1^2 - 0^2 = 1$.

Summary Table: Key Formulas

Concept	Formula	Description
Norm of $\vec{x}$ in $\mathbb{R}^n$	$\|\vec{x}\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}$	Length of a vector
Dot Product	$\vec{x} \cdot \vec{x} = \|\vec{x}\|^2$	Square of the norm
Unit Vector	$\hat{x} = \dfrac{\vec{x}}{\|\vec{x}\|}$	Vector of length 1 in the direction of $\vec{x}$
Determinant of $2 \times 2$ Matrix	$\det(A) = a_{11}a_{22} - a_{12}a_{21}$	Area scaling factor for linear transformation
Gram Determinant	$G(\vec{a}, \vec{b}) = \|\vec{a}\|^2 \|\vec{b}\|^2 - (\vec{a} \cdot \vec{b})^2$	Area/volume spanned by vectors