/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 Find the vector \(\mathbf{v}\) o... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 35

Find the vector \(\mathbf{v}\) of length 6 that has the same direction as the unit vector \(\langle 1 / 2, \sqrt{3} / 2\rangle\)

Short Answer

Expert verified
Answer: The vector is \(\langle 3, 3\sqrt{3} \rangle\).

Step by step solution

01

Identify the given unit vector

The given unit vector, \(\mathbf{u}\), is \(\langle 1 / 2, \sqrt{3} / 2 \rangle\).
02

Determine the scale factor for the desired vector length

We want to find a vector \(\mathbf{v}\) of length 6 that has the same direction as the unit vector \(\mathbf{u}\). To do this, we need to scale the unit vector by a factor of 6.
03

Multiply each component of the unit vector by the scale factor

To find the vector \(\mathbf{v}\) of length 6 in the same direction as the unit vector \(\mathbf{u}\), we multiply each component of the unit vector by the scale factor: \(\begin{bmatrix} 1/2 \\ \sqrt{3}/2 \end{bmatrix} \cdot 6 = \begin{bmatrix} 6/2 \\ 6\sqrt{3}/2 \end{bmatrix}\).
04

Simplify the resulting vector components

The resulting vector \(\mathbf{v}\) is: \(\begin{bmatrix} 6/2 \\ 6\sqrt{3}/2 \end{bmatrix} = \begin{bmatrix} 3 \\ 3\sqrt{3} \end{bmatrix}\). So, the vector \(\mathbf{v}\) of length 6 in the same direction as the unit vector \(\langle 1 / 2, \sqrt{3} / 2\rangle\) is \(\mathbf{v} = \langle 3, 3\sqrt{3} \rangle\).

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Key Concepts

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

unit vector
A unit vector is a vector that has a length or magnitude of exactly 1. Unit vectors are helpful when you only care about the direction of a vector and not its size.
You can identify a unit vector by calculating its magnitude using the formula:
  • The components of the vector are squared.
  • The squares are summed.
  • The square root of the sum gives the magnitude.
If this magnitude equals 1, you have a unit vector. Because of their consistent length, unit vectors make it easier to find other vectors with different lengths in the same direction.
Unit vectors are often noted with a small hat symbol like \(\hat{i}, \hat{j}, \hat{k}\) in three-dimensional space.
vector direction
The direction of a vector refers to the orientation it points to in space. It is the primary role that the unit vector captures, while ignoring the vector's length.
In a physical or numerical sense, the direction is given by the angle or set of angles the vector makes with reference points like the x-axis, y-axis, and so on.
To preserve the direction but change the length, you scale the unit vector that represents this direction by multiplying it by a desired magnitude. For example, if we take the unit vector \(\langle 1/2, \sqrt{3}/2 \rangle\), and multiply it by 6, the resulting vector will point in the same direction, but will be six times as long.
vector length
The length of a vector, also known as its magnitude, is a measure of how long the vector is in space. It鈥檚 calculated using the components of the vector with the formula: \[ \text{Length} = \sqrt{x^2 + y^2} \] where \(x\) and \(y\) are the components of a 2D vector.
For a given unit vector, you can find a vector of desired length by simply scaling the unit vector. This is done by multiplying each component of the unit vector by the length you want the final vector to have.
So, for a unit vector \(\langle 1/2, \sqrt{3}/2 \rangle\), if you want a length of 6, you scale it by dividing each component by its current magnitude, then multiplying by 6, resulting in \(\langle 3, 3\sqrt{3} \rangle\).
This retains the original direction but changes its length to the one you need.

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Most popular questions from this chapter

Linear combinations A sum of scalar multiples of two or more vectors (such as \(c_{1} \mathbf{u}+c_{2} \mathbf{v}+c_{3} \mathbf{w},\) where \(c_{i}\) are scalars) is called a linear combination of the vectors. Let \(\mathbf{i}=\langle 1,0\rangle, \mathbf{j}=\langle 0,1\rangle\) \(\mathbf{u}=\langle 1,1\rangle,\) and \(\mathbf{v}=\langle-1,1\rangle\) For arbitrary real numbers \(a\) and \(b\), express \(\langle a, b\rangle\) as a linear combination of \(\mathbf{u}\) and \(\mathbf{v}\)

Equal vectors For the points \(A(3,4), B(6,10), C(a+2, b+5)\) and \(D(b+4, a-2),\) find the values of \(a\) and \(b\) such that \( {A B}= {C D}\)

Vector equations Use the properties of vectors to solve the following equations for the unknown vector \(\mathbf{x}=\langle a, b\rangle .\) Let \(\mathbf{u}=\langle 2,-3\rangle\) and \(\mathbf{v}=\langle-4,1\rangle\). $$10 x=u$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The line \(r=(3,-1,4)+t\langle 6,-2,8\rangle\) passes through the origin. b. Any two nonparallel lines in \(\mathbb{R}^{3}\) intersect. c. The plane \(x+y+z=0\) and the line \(x=t, y=t, z=t\) are parallel. d. The vector equations \(r=\langle 1,2,3\rangle+t\langle 1,1,1\rangle\) and \(\mathbf{R}=(1,2,3)+t(-2,-2,-2)\) describe the same line. e. The equations \(x+y-z=1\) and \(-x-y+z=1\) describe the same plane. I. Any two distinct lines in \(\mathbb{R}^{3}\) determine a unique plane. g. The vector \langle-1,-5,7\rangle is perpendicular to both the line \(x=1+5 t, y=3-t, z=1\) and the line \(x=7 t, y=3\)

Determine whether the points \(P\). \(Q\). and \(R\) are collinear (lie on a line) by comparing \(P Q\) and \(P R .\) If the points are collinear, determine which point lies between the other two points. a. \(P(1,6,-5), Q(2,5,-3), R(4,3,1)\) b. \(P(1,5,7), Q(5,13,-1), R(0,3,9)\) c. \(P(1,2,3), Q(2,-3,6), R(3,-1,9)\) d. \(P(9,5,1), Q(11,18,4), R(6,3,0)\)
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