Problem 3 - Algebra and Trigonometry

Question

A vector \(\mathbf{u}\) for which \(\|\mathbf{u}\|=1\) is called a(n) _____ vector.

Step-by-Step Solution

Answer

Unit vector

Step 1: Understand the Problem

Determine the concept that describes a vector with a magnitude (or norm) of 1.

Step 2: Recall Key Definitions

Recall that a vector whose norm (or length) is 1 is called a 'unit vector'. Mathematically, if \(\|\mathbf{u}\| = 1\)\, then \(\mathbf{u}\) is a unit vector.

Step 3: Conclusion

Based on the definition, conclude that a vector with a norm of 1 is called a unit vector.

Key Concepts

vector magnitude

A vector's magnitude provides information about the length or size of the vector. It is sometimes also referred to as the 'length' or 'norm' of the vector.

For a vector \(\boldsymbol{v} \) in a 2-dimensional or 3-dimensional space, its magnitude is computed using the Pythagorean Theorem. Let's say \(\boldsymbol{v} = (v_1, v_2)\) in 2D or \(\boldsymbol{v} = (v_1, v_2, v_3)\) in 3D. The magnitude \(\boldsymbol{\text{of v}}\) can be calculated as:

\[\text{In 2D:} \|\boldsymbol{v}\| = \sqrt{v_1^2 + v_2^2}\ \] \[ \text{In 3D:} \|\boldsymbol{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}\ \]

The result will always be a non-negative number.

Understanding vector magnitude is crucial in fields like physics, engineering, and computer graphics to measure distances and define directions.

norm of a vector

The norm of a vector usually refers to its magnitude or length in a given vector space. It is a function that assigns a strictly positive length or size to each vector in the space, except for the zero vector, which has a norm of zero.

There are different types of norms, but the most common one is the Euclidean norm, represented as \(\boldsymbol{\text{\|v\|}}\). For a vector \(\boldsymbol{v} = (v_1, v_2, ..., v_n)\):

\[\text{Euclidean norm:} \|v\| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2} \]

Other types of norms include the Manhattan norm (or L1 norm), which is the sum of the absolute values of the components:

\[\boldsymbol{\text{Manhattan norm:}} \|v\|_1 = |v_1| + |v_2| + ... + |v_n| \]

And the infinity norm, which is the maximum absolute value among the vector's components:

\[\boldsymbol{\text{Infinity norm:}} \|v\|_\text{∞} = \text{max}(|v_1|, |v_2|, ..., |v_n|) \]

The Euclidean norm is the most commonly used and referred to simply as the 'norm' or 'magnitude' of the vector.

unit vector

A unit vector is a vector that has a magnitude (or norm) of exactly 1. Unit vectors are often used to represent direction, as they retain the vector's direction but have a standard length.

If you have a vector \(\boldsymbol{v}\) that is not a unit vector, you can convert it into one by dividing the vector by its magnitude:

\[\text{Unit vector} \ \boldsymbol{\text{u}} = \ \frac{\boldsymbol{v}}{\boldsymbol{\text{\|v\|}}} \]

Here, \(\boldsymbol{v} \) is your original vector, and \(\boldsymbol{\text{\|v\|}} \) is its magnitude.

For example, if \(\boldsymbol{v} = (3, 4) \), the magnitude is \(\boldsymbol{\text{\|v\|}} = 5 \), and the unit vector \(\boldsymbol{u} \) is:

\[\boldsymbol{u} = \frac{(3, 4)}{5} = (0.6, 0.8) \]

Unit vectors are denoted with a hat, such as \(\boldsymbol{\text{\hat{i}}} \) and \(\boldsymbol{\text{\hat{j}}} \) in two dimensions, representing the x and y directions, respectively.

Understanding unit vectors is essential because they simplify mathematical computations and are used in various fields like physics, computer graphics, and navigation.