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Chapter 9: Problem 11

Normalize \([1,3,-1]\).

Short Answer

Expert verified
The normalized vector is \(\left( \frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}} \right).\)

Step by step solution

01

Calculate the Magnitude of the Vector

The magnitude of a vector \([x, y, z]\) is calculated using the formula \( ext{magnitude} = \sqrt{x^2 + y^2 + z^2}\). For the vector \([1, 3, -1]\), calculate the magnitude as follows: \[\sqrt{1^2 + 3^2 + (-1)^2} = \sqrt{1 + 9 + 1} = \sqrt{11}.\] Thus, the magnitude is \(\sqrt{11}\).
02

Divide Each Component by the Magnitude

Normalize the vector by dividing each component of the original vector by its magnitude. Using the calculated magnitude \(\sqrt{11}\), the normalized vector is: \[\left( \frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}} \right).\]
03

Simplify the Components

To simplify, each component can be expressed as a rational number under the square root of 11. Thus, the final normalized vector is: \[\left( \frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}} \right).\] This vector has a magnitude of 1, confirming it is normalized.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnitude of a Vector
In the world of vectors, understanding the magnitude is crucial. The magnitude of a vector is similar to the length of a line segment in geometry. It provides the vector's size but tells us nothing about its direction. To calculate the magnitude, we use the formula for a vector with components \([x, y, z]\): \[\sqrt{x^2 + y^2 + z^2}\]. This involves squaring each component, adding them up, and then taking the square root. For example, with the vector \([1, 3, -1]\), you perform the following:
  • Square each number: \(1^2 = 1\), \(3^2 = 9\), and \((-1)^2 = 1\).
  • Sum these squares: \(1 + 9 + 1 = 11\).
  • Find the square root of that sum: \(\sqrt{11}\).
This number, \(\sqrt{11}\), is the magnitude. It is a critical first step in normalizing a vector, setting the stage for transforming it into a vector of magnitude one.
Normalization Process
Normalization in vectors is much like putting a hat on a vector. It transforms a given vector into a unit vector, which has a magnitude of exactly one. This process is helpful in various applications, such as simplifying vector computations, ensuring consistent vector lengths, and finding directions. There are generally three steps to normalize a vector:
  • Calculate the magnitude: As discussed earlier, determine the magnitude of the vector.
  • Divide each component by the magnitude: Each component of the vector is divided by the magnitude. This scales the vector down to a unit length. For example, with the vector \([1, 3, -1]\) and its magnitude \(\sqrt{11}\), you'd divide each component: \(\frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}}\).
  • Verify the result: Ensure the resultant vector's magnitude is exactly one, confirming a successful normalization.
This process is a pivotal concept in linear algebra and vectors, ensuring vectors are readily usable and comparable in multiple dimensions.
Vector Components Normalization
The final step in the normalization process involves dividing each component of a vector by its magnitude. This step ensures each individual component is adjusted to fit into a unit vector, serving the purpose of direction rather than size.Here's a simple breakdown of how to normalize each component:
  • Take the original vector components. For the vector \([1, 3, -1]\), these are 1, 3, and -1.
  • Divide each by the magnitude \(\sqrt{11}\):
    • First component: \(\frac{1}{\sqrt{11}}\)
    • Second component: \(\frac{3}{\sqrt{11}}\)
    • Third component: \(\frac{-1}{\sqrt{11}}\)
  • Resulting vector is now the normalized vector: \(\left( \frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}} \right)\).
By completing this step, you ensure that the resulting vector now maintains its original direction but has been scaled down to a size (magnitude) of one. This makes it extremely useful in fields that require direction-based calculations, such as physics and computer graphics.

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Most popular questions from this chapter

Find the parametric equation of the line in \(x-y-z\) space that goes through the indicated point in the direction of the indicated vector. $$(-1,3,-2),\left[\begin{array}{r}-1 \\ -2 \\ 4\end{array}\right]$$

Let \(A=(1,3,-2)\) and \(B=(0,-1,-1)\). Find the vector representation of \(\overrightarrow{A B}\).

Let \(\mathbf{x}=[1,0,-1]^{\prime}\) and \(\mathbf{y}=[-2,1,0]^{\prime}\). (a) Find \(\mathbf{x}+\mathbf{y}\). (b) Find \(2 \mathbf{x}\). (c) Find \(-3 \mathbf{y}\).

In Problems \(57-60\), find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{ll}4 & 0 \\ 0 & 3\end{array}\right]$$

Assume the given Leslie matrix L. Determine the number of age classes in the population, the fraction of one-year-olds present at time \(t\) that survive to time \(t+1\), and the average number of female offspring of a two-year-old female. $$L=\left[\begin{array}{lll}0 & 5 & 0 \\ 0.8 & 0 & 0 \\ 0 & 0.4 & 0\end{array}\right]$$
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