Question

Compute \(\|\mathbf{v}\|\) if \(\mathbf{v}\) equals: a. \(\left[\begin{array}{r}2 \\ -1 \\ 2\end{array}\right]\) b. \(\left[\begin{array}{r}1 \\ -1 \\ 2\end{array}\right]\) c. \(\left[\begin{array}{r}1 \\ 0 \\ -1\end{array}\right]\) d. \(\left[\begin{array}{r}-1 \\ 0 \\ 2\end{array}\right]\) e. \(2\left[\begin{array}{r}1 \\ -1 \\ 2\end{array}\right]\) f. \(-3\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]\)

Short answer

a: 3, b: \( \sqrt{6} \), c: \( \sqrt{2} \), d: \( \sqrt{5} \), e: \( 2\sqrt{6} \), f: \( 3\sqrt{6} \).

Step-by-step solution

Understanding Vector Norm

The norm of a vector, denoted by \( \|\mathbf{v}\| \), is a measure of the vector's magnitude. For a vector \( \mathbf{v} = \left[ \begin{array}{c} a \ b \ c \end{array} \right] \), the norm is calculated using the formula: \( \|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2} \). This is also known as the Euclidean norm or the magnitude of the vector.

Calculating Norm for Vector a

Given \( \mathbf{v}_a = \left[ \begin{array}{r} 2 \ -1 \ 2 \end{array} \right] \), the norm \( \|\mathbf{v}_a\| \) is calculated as follows: \( \|\mathbf{v}_a\| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \).

Calculating Norm for Vector b

For \( \mathbf{v}_b = \left[ \begin{array}{r} 1 \ -1 \ 2 \end{array} \right] \), compute the norm as \( \|\mathbf{v}_b\| = \sqrt{1^2 + (-1)^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \).

Calculating Norm for Vector c

For \( \mathbf{v}_c = \left[ \begin{array}{r} 1 \ 0 \ -1 \end{array} \right] \), compute the norm as \( \|\mathbf{v}_c\| = \sqrt{1^2 + 0^2 + (-1)^2} = \sqrt{1 + 0 + 1} = \sqrt{2} \).

Calculating Norm for Vector d

For \( \mathbf{v}_d = \left[ \begin{array}{r} -1 \ 0 \ 2 \end{array} \right] \), compute the norm as \( \|\mathbf{v}_d\| = \sqrt{(-1)^2 + 0^2 + 2^2} = \sqrt{1 + 0 + 4} = \sqrt{5} \).

Scaling Vector e and Calculating Norm

Given \( \mathbf{v}_e = 2\left[ \begin{array}{r} 1 \ -1 \ 2 \end{array} \right] = \left[ \begin{array}{r} 2 \ -2 \ 4 \end{array} \right] \), compute the norm \( \|\mathbf{v}_e\| = \sqrt{2^2 + (-2)^2 + 4^2} = \sqrt{4 + 4 + 16} = \sqrt{24} = 2\sqrt{6} \).

Scaling Vector f and Calculating Norm

Given \( \mathbf{v}_f = -3\left[ \begin{array}{r} 1 \ 1 \ 2 \end{array} \right] = \left[ \begin{array}{r} -3 \ -3 \ -6 \end{array} \right] \), compute the norm \( \|\mathbf{v}_f\| = \sqrt{(-3)^2 + (-3)^2 + (-6)^2} = \sqrt{9 + 9 + 36} = \sqrt{54} = 3\sqrt{6} \).

Key concepts

Euclidean Norm

The Euclidean Norm, often simply called the "norm," is a key concept in linear algebra that allows us to measure the length (or magnitude) of a vector. To understand this better, think of a vector as a point in space. The Euclidean Norm gives us the straight-line distance from the origin of the space, which we can imagine as measuring a physical object from end to end.
The formula to calculate the Euclidean Norm for a vector \(\mathbf{v} = \left[ \begin{array}{c} a \ b \ c \end{array} \right]\) is \(\|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2}\). This is the same formula used to calculate the distance in three-dimensional space using the Pythagorean theorem.
When you apply this to different vectors by plugging in their components into this formula, you get the specific length of each vector. Understanding and finding the Euclidean Norm is essential when working with vectors in various applications, such as physics and data science.

Vector Magnitude

Vector Magnitude is essentially another name for the Euclidean Norm. It's important because it helps us understand the vector's size without considering its direction.
Just like how weight scales work, the magnitude of a vector tells us how much "power" or "strength" the vector carries. Whether a vector points upward, downward, or sideways, its magnitude remains constant.
To find the magnitude, you simply follow the same formula used for the Euclidean Norm. In linear algebra applications, knowing the magnitude of a vector is crucial when comparing vectors or determining similarities between them.

Linear Algebra

Linear Algebra is the branch of mathematics that deals with vectors, among other things. It provides the framework for working with vectors, matrices, and linear transformations. Vectors are fundamental building blocks in linear algebra, and understanding vector norms is crucial.
In linear algebra, norms like the Euclidean Norm are utilized to analyze vector spaces' properties. This calculation helps in measuring distances and angles between vectors, which is foundational in fields such as 3D graphics, engineering, and machine learning.
Grasping linear algebra basics, including vector operations and norms, is vital for students and professionals dealing with data or models that rely on mathematical abstractions.

Vector Operations

Vector Operations include a variety of mathematical techniques, such as addition, subtraction, scalar multiplication, and finding the norm. Each operation serves a unique purpose and allows us to manipulate vectors to fit our needs.
For instance, scalar multiplication involves multiplying a vector by a number, which scales the vector—making it longer or shorter but not changing its direction. In the exercise above, we see this with vectors \(\mathbf{v}_e\) and \(\mathbf{v}_f\).
Understanding these operations is crucial in fields ranging from physics, where vectors can represent forces, to computer science, where they can represent data points. Mastery of vector operations helps in performing complex calculations easily and accurately.