Solved: In Exercises 6368, find and such that where and w 1, 1.v 3, 3

In Exercises 1–4, (a) find the component form of the vector and (b) sketch the vector v with its initial point at the origin.

In Exercises 1–4, (a) find the component form of the vector and (b) sketch the vector v with its initial point at the origin.

In Exercises 1–4, (a) find the component form of the vector and (b) sketch the vector v with its initial point at the origin.

In Exercises 1–4, (a) find the component form of the vector and (b) sketch the vector v with its initial point at the origin.

In Exercises 5–8, find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. u: (3, 2), (5, 6) v: (1, 4), (3, 8)

In Exercises 5–8, find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. u: (-4, 0), (1, 8) v: (2, -1), (7, 7)

In Exercises 5–8, find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. u: (0, 3), (6, 2) v: (3, 10), (9, 5)

In Exercises 5–8, find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. u: (-4, -1), (11, -4) v: (10, 13), (25, 10)

In Exercises 9–16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line segment, (b) write the vector in component form, (c) write the vector as the linear combination of the standard unit vectors i and j, and (d) sketch the vector with its initial point at the origin. Initial Point Terminal Point (2, 0) (5, 5)

In Exercises 9–16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line segment, (b) write the vector in component form, (c) write the vector as the linear combination of the standard unit vectors i and j, and (d) sketch the vector with its initial point at the origin. Initial Point Terminal Point (4, -6) (3, 6)

In Exercises 9–16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line segment, (b) write the vector in component form, (c) write the vector as the linear combination of the standard unit vectors i and j, and (d) sketch the vector with its initial point at the origin. Initial Point Terminal Point (8, 3) (6, -1)

In Exercises 9–16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line segment, (b) write the vector in component form, (c) write the vector as the linear combination of the standard unit vectors i and j, and (d) sketch the vector with its initial point at the origin. Initial Point Terminal Point (0, -4) (-5, -1)

In Exercises 9–16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line segment, (b) write the vector in component form, (c) write the vector as the linear combination of the standard unit vectors i and j, and (d) sketch the vector with its initial point at the origin. Initial Point Terminal Point (6, 2) (6, 6)

In Exercises 9–16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line segment, (b) write the vector in component form, (c) write the vector as the linear combination of the standard unit vectors i and j, and (d) sketch the vector with its initial point at the origin. Initial Point Terminal Point (6, 2) (-3, -1)

In Exercises 9–16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line segment, (b) write the vector in component form, (c) write the vector as the linear combination of the standard unit vectors i and j, and (d) sketch the vector with its initial point at the origin. Initial Point Terminal Point \(\left(\frac{3}{2},\frac{4}{3}\right)\) \(\left(\frac{1}{2}, 3\right)\) Text Transcription: (3/2,4/3) (1/2,3)

In Exercises 9–16, the initial and terminal points of a vector v are given. (a) Sketch the given directed line segment, (b) write the vector in component form, (c) write the vector as the linear combination of the standard unit vectors i and j, and (d) sketch the vector with its initial point at the origin. Initial Point Terminal Point (0.12, 0.60) (0.84, 1.25)

In Exercises 17 and 18, sketch each scalar multiple of v. \(\mathbf{v}=\langle 3,5\rangle\) (a) 2v (b) -3v (c) \(\frac{7}{2} \mathbf{v}\) (d) \(\frac{2}{3} \mathbf{v}\) Text Transcription: v=<3,5> 7/2 v 2/3 v

In Exercises 17 and 18, sketch each scalar multiple of v. \(\mathbf{v}=\langle-2,3\rangle\) (a) 4v (b) \(-\frac{1}{2} \mathbf{v}\) (c) 0v (d) -6v Text Transcription: v=<-2,3> -1/2 v

In Exercises 19–22, use the figure to sketch a graph of the vector. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. -u

In Exercises 19–22, use the figure to sketch a graph of the vector. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 2u

In Exercises 19–22, use the figure to sketch a graph of the vector. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. u - v

In Exercises 19–22, use the figure to sketch a graph of the vector. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. u + 2v

In Exercises 23 and 24, find (a) \(\frac{2}{3}\frac{2}{3} \mathbf{u}\), (b) v - u, and (c) 2u + 5v. \(\mathbf{u}=\langle 4,9\rangle\) \(\mathbf{v}=\langle 2,-5\rangle\) Text Transcription: 2/3 u u=<4,9> v=<2,-5>

In Exercises 23 and 24, find (a) \(\frac{2}{3}\frac{2}{3} \mathbf{u}\), (b) v - u, and (c) 2u + 5v. \(\mathbf{u}=\langle-3,-8\rangle\) \(\mathbf{v}=\langle 8,25\rangle\) Text Transcription: 2/3 u u=<-3,-8> v=<8,25>

In Exercises 25-28, find the vector v where \(\mathbf{u}=\langle 2,-1\rangle\) and \(w=\langle 1,2\rangle\). Illustrate the vector operations geometrically. \(\mathbf{v}=\frac{3}{2} \mathbf{u}\) Text Transcription: u=<2, -1> w=<1, -2> v=3/2 u

In Exercises 25-28, find the vector v where \(\mathbf{u}=\langle 2,-1\rangle\) and \(w=\langle 1,2\rangle\). Illustrate the vector operations geometrically. v = u + w Text Transcription: u=<2, -1> w=<1, -2>

In Exercises 25-28, find the vector v where \(\mathbf{u}=\langle 2,-1\rangle\) and \(w=\langle 1,2\rangle\). Illustrate the vector operations geometrically. v = u + 2w Text Transcription: u=<2, -1> w=<1, -2>

In Exercises 25-28, find the vector v where \(\mathbf{u}=\langle 2,-1\rangle\) and \(w=\langle 1,2\rangle\). Illustrate the vector operations geometrically. v = 5u - 3w Text Transcription: u=<2, -1> w=<1, -2>

In Exercises 29 and 30, the vector v and its initial point are given. Find the terminal point. \(\mathbf{v}=\langle-1,3\rangle\); Initial point: (4, 2) Text Transcription: v=<-1, 3>

In Exercises 29 and 30, the vector v and its initial point are given. Find the terminal point. \(\mathbf{v}=\langle 4,-9\rangle\); Initial point: (5, 3) Transcription: v=<4, -9>

In Exercises 31-36, find the magnitude of v. v = 7i

In Exercises 31-36, find the magnitude of v. v = -3i

In Exercises 31-36, find the magnitude of v. \(\mathbf{v}=\langle 4,3\rangle\) Text Transcription: v = <4, 3>

In Exercises 31-36, find the magnitude of v. \(\mathbf{v}=\langle 12,-5\rangle\) Text Transcription: v = <12, -5>

In Exercises 31-36, find the magnitude of v. v= 6i - 5j

In Exercises 31-36, find the magnitude of v. v = -10i + 3j

In Exercises 37-40, find the unit vector in the direction of v and verify that it has length 1. \(\mathbf{v}=\langle 3,12\rangle\) Text Transcription: v = <3, 12>

In Exercises 37-40, find the unit vector in the direction of v and verify that it has length 1. \(\mathbf{v}=\langle -5,15\rangle\) Text Transcription: v = <-5, 15>

In Exercises 37-40, find the unit vector in the direction of v and verify that it has length 1. \(V=\left\langle\frac{3}{2}, \frac{5}{2}\right\rangle\) Text Transcription: v = <3/2, 5/2>

In Exercises 37-40, find the unit vector in the direction of v and verify that it has length 1. \(\mathbf{v}=\langle -6.2, 3.4\rangle\) Text Transcription: v = <-6.2, 3.4>

In Exercises 41-44, find the following. (a) ||u|| (b) ||v|| (c) ||u + v|| (d) \(\left\|\frac{\mathbf{u}}{\|\mathbf{u}\|}\right\|\) (e) \(\left\|\frac{\mathbf{V}}{\|\mathbf{V}\|}\right\|\) (f) \(\left\|\frac{\mathbf{u}+\mathbf{v}}{\|\mathbf{u}+\mathbf{v}\|}\right\|\) \(\begin{array}{l}\mathbf{u}=\langle 1,-1\rangle \\\mathbf{v}=\langle-1,2\rangle\end{array}\) Text Transcription: ||u/||u|||| ||v/||v|||| ||u+v/||u+v|||| u=<1, -1> v=<-1,2>

In Exercises 41-44, find the following. (a) ||u|| (b) ||v|| (c) ||u + v|| (d) \(\left\|\frac{\mathbf{u}}{\|\mathbf{u}\|}\right\|\) (e) \(\left\|\frac{\mathbf{V}}{\|\mathbf{V}\|}\right\|\) (f) \(\left\|\frac{\mathbf{u}+\mathbf{v}}{\|\mathbf{u}+\mathbf{v}\|}\right\|\) \(\begin{array}{l}\mathbf{u}=\langle 0,1\rangle \\\mathbf{v}=\langle 3,-3\rangle\end{array}\) Text Transcription: ||u/||u|||| ||v/||v|||| ||u+v/||u+v|||| u=<0, 1> v=<3,-3>

In Exercises 41-44, find the following. (a) ||u|| (b) ||v|| (c) ||u + v|| (d) \(\left\|\frac{\mathbf{u}}{\|\mathbf{u}\|}\right\|\) (e) \(\left\|\frac{\mathbf{V}}{\|\mathbf{V}\|}\right\|\) (f) \(\left\|\frac{\mathbf{u}+\mathbf{v}}{\|\mathbf{u}+\mathbf{v}\|}\right\|\) \(\begin{array}{l}\mathbf{u}=\left\langle 1, \frac{1}{2}\right\rangle \\\mathbf{v}=\langle 2,3\rangle\end{array}\) Text Transcription: ||u/||u|||| ||v/||v|||| ||u+v/||u+v|||| u=<1, 1/2> v=<2,3>

In Exercises 41-44, find the following. (a) ||u|| (b) ||v|| (c) ||u + v|| (d) \(\left\|\frac{\mathbf{u}}{\|\mathbf{u}\|}\right\|\) (e) \(\left\|\frac{\mathbf{V}}{\|\mathbf{V}\|}\right\|\) (f) \(\left\|\frac{\mathbf{u}+\mathbf{v}}{\|\mathbf{u}+\mathbf{v}\|}\right\|\) \(\begin{array}{l}\mathbf{u}=\langle 2,-4\rangle \\\mathbf{v}=\langle 5,5\rangle\end{array}\) Text Transcription: ||u/||u|||| ||v/||v|||| ||u+v/||u+v|||| u=<2, -4> v=<5,5>

In Exercises 45 and 46, sketch a graph of u, v, and u + v. Then demonstrate the triangle inequality using the vectors u and v. \(\mathbf{u}=\langle 2,1\rangle, \quad \mathbf{v}=\langle 5,4\rangle\) Text Transcription: u=<2,1>,v=<5,4>

In Exercises 45 and 46, sketch a graph of u, v, and u + v. Then demonstrate the triangle inequality using the vectors u and v. \(\mathbf{u}=\langle-3,2\rangle, \quad \mathbf{v}=\langle 1,-2\rangle\) Text Transcription: u=<-3,2>,v=<1,-2>

In Exercises 47-50, find the vector v with the given magnitude and the same direction as u. Magnitude Direction ||v|| = 6 \(\mathbf{u}=\langle 0,3\rangle\) Text Transcription: u=<0,3>

In Exercises 47-50, find the vector v with the given magnitude and the same direction as u. Magnitude Direction ||v|| = 4 \(\mathbf{u}=\langle 1,1\rangle\) Text Transcription: u=<1,1>

In Exercises 47-50, find the vector v with the given magnitude and the same direction as u. Magnitude Direction ||v|| = 5 \(\mathbf{u}=\langle -1,2\rangle\) Text Transcription: u=<-1,2>

In Exercises 47-50, find the vector v with the given magnitude and the same direction as u. Magnitude Direction ||v|| = 2 \(\mathbf{u}=\langle\sqrt{3}, 3\rangle\) Text Transcription: u=<sqrt 3,3>

In Exercises 51-54, find the component form of v given its magnitude and the angle it makes with the positive x-axis. \(\|\mathbf{v}\|=3, \quad \theta=0^{\circ}\) Text Transcription: ||v||=3, theta=0 degree

In Exercises 51-54, find the component form of v given its magnitude and the angle it makes with the positive x-axis. \(\|\mathbf{v}\|=5, \quad \theta=120^{\circ}\) Text Transcription: ||v||=5, theta=120 degrees

In Exercises 51-54, find the component form of v given its magnitude and the angle it makes with the positive x-axis. \(\|\mathbf{v}\|=2, \quad \theta=150^{\circ}\) Text Transcription: ||v||=2, theta=150 degrees

In Exercises 51-54, find the component form of v given its magnitude and the angle it makes with the positive x-axis. \(\|\mathbf{v}\|=4, \quad \theta=3.5^{\circ}\) Text Transcription: ||v||=4, theta=3.5 degrees

In Exercises 55-58, find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. \(\begin{array}{ll}\|\mathbf{u}\|=1, & \theta_{\mathbf{u}}=0^{\circ} \\\|\mathbf{v}\|=3, & \theta_{\mathbf{v}}=45^{\circ}\end{array}\) Text Transcription: ||u||=1, theta_u=0 degree ||v||=3, theta_v=45 degrees

In Exercises 55-58, find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. \(\begin{array}{ll}\|\mathbf{u}\|=4, & \theta_{\mathbf{u}}=0^{\circ} \\\|\mathbf{v}\|=2, & \theta_{\mathbf{v}}=60^{\circ}\end{array}\) Text Transcription: ||u||=4, theta_u=0 degree ||v||=2, theta_v=60 degrees

In Exercises 55-58, find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. \(\begin{array}{ll}\|\mathbf{u}\|=2, & \theta_{\mathbf{u}}=4 \\\|\mathbf{v}\|=1, & \theta_{\mathbf{v}}=2\end{array}\) Text Transcription: ||u||=2, theta_u=4 ||v||=1, theta_v=2

In Exercises 55-58, find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. \(\begin{array}{ll}\|\mathbf{u}\|=5, & \theta_{\mathbf{u}}=-0.5 \\\|\mathbf{v}\|=5, & \theta_{\mathbf{v}}=0.5\end{array}\) Text Transcription: ||u||=5, theta_u=-0.5 ||v||=5, theta_v=0.5

In your own words, state the difference between a scalar and a vector. Give examples of each.

Give geometric descriptions of the operations of addition of vectors and multiplication of a vector by a scalar.

Identify the quantity as a scalar or as a vector. Explain your reasoning. (a) The muzzle velocity of a gun (b) The price of a company's stock

Identify the quantity as a scalar or as a vector. Explain your reasoning. (a) The air temperature in a room (b) The weight of a car

In Exercises 63-68, find a and b such that v = au + bw, where \(\mathbf{u}=\langle 1,2\rangle\) and \(\mathbf{w}=\langle 1, -1\rangle\) \(\mathbf{v}=\langle 2, 1\rangle\) Text Transcription: u=<1, 2> w=<1, -1>61 v=<2, 1>

In Exercises 63-68, find a and b such that v = au + bw, where \(\mathbf{u}=\langle 1,2\rangle\) and \(\mathbf{w}=\langle 1, -1\rangle\) \(\mathbf{v}=\langle 0, 3\rangle\) Text Transcription: u=<1, 2> w=<1, -1>61 v=<0, 3>

In Exercises 63-68, find a and b such that v = au + bw, where \(\mathbf{u}=\langle 1,2\rangle\) and \(\mathbf{w}=\langle 1, -1\rangle\) \(\mathbf{v}=\langle 3, 0\rangle\) Text Transcription: u=<1, 2> w=<1, -1>61 v= <3, 0>

In Exercises 63-68, find a and b such that v = au + bw, where \(\mathbf{u}=\langle 1,2\rangle\) and \(\mathbf{w}=\langle 1, -1\rangle\) \(\mathbf{v}=\langle 3, 3\rangle\) Text Transcription: u=<1, 2> w=<1, -1>61 v= <3, 3>

In Exercises 63-68, find a and b such that v = au + bw, where \(\mathbf{u}=\langle 1,2\rangle\) and \(\mathbf{w}=\langle 1, -1\rangle\) \(\mathbf{v}=\langle 1, 1\rangle\) Text Transcription: u=<1, 2> w=<1, -1>61 v= <1, 1>

In Exercises 63-68, find a and b such that v = au + bw, where \(\mathbf{u}=\langle 1,2\rangle\) and \(\mathbf{w}=\langle 1, -1\rangle\) \(\mathbf{v}=\langle -1, 7\rangle\) Text Transcription: u=<1, 2> w=<1, -1>61 v= <-1, 7>

In Exercises 69-74, find a unit vector (a) parallel to and (b) perpendicular to the graph of f at the given point. Then sketch the graph of f and sketch the vectors at the given point. Function Point \(f(x)=x^{2}\) (3, 9) Text Transcription: f(x)=x^2

In Exercises 69-74, find a unit vector (a) parallel to and (b) perpendicular to the graph of f at the given point. Then sketch the graph of f and sketch the vectors at the given point. Function Point \(f(x)=-x^{2}+5\) (1, 4) Text Transcription: f(x)=x^2+5

In Exercises 69-74, find a unit vector (a) parallel to and (b) perpendicular to the graph of f at the given point. Then sketch the graph of f and sketch the vectors at the given point. Function Point \(f(x)=x^{3}\) (1, 1) Text Transcription: f(x)=x^3

In Exercises 69-74, find a unit vector (a) parallel to and (b) perpendicular to the graph of f at the given point. Then sketch the graph of f and sketch the vectors at the given point. Function Point \(f(x)=x^{3}\) (-2, -8) Text Transcription: f(x)=x^3

In Exercises 69-74, find a unit vector (a) parallel to and (b) perpendicular to the graph of f at the given point. Then sketch the graph of f and sketch the vectors at the given point. Function Point \(f(x)=\sqrt{25-x^{2}}\) (3, 4) Text Transcription: f(x)=sqrt 25-x^2

In Exercises 69-74, find a unit vector (a) parallel to and (b) perpendicular to the graph of f at the given point. Then sketch the graph of f and sketch the vectors at the given point. Function Point \(f(x)=tan x\) (\(\frac{\pi}{4}\), 1) Text Transcription: f(x)=tan x pi/4

In Exercises 75 and 76, find the component form of v given the magnitudes of u and u + v and the angles that u and u + v make with the positive x-axis. \(\begin{array}{l}\|\mathbf{u}\|=1, \theta=45^{\circ} \\\|\mathbf{u}+\mathbf{v}\|=\sqrt{2}, \theta=90^{\circ}\end{array}\) Text Transcription: ||u||=1, theta = 45 degree ||u+v||=sqrt 2, theta = 90 degree

In Exercises 75 and 76, find the component form of v given the magnitudes of u and u + v and the angles that u and u + v make with the positive x-axis. \(\begin{array}{l}\|\mathbf{u}\|=4, \theta=30^{\circ} \\\|\mathbf{u}+\mathbf{v}\|=6, \theta=120^{\circ}\end{array}\) Text Transcription: ||u||=4, theta = 30 degree ||u+v||=6, theta = 120 degree

Programming You are given the magnitudes of u and v and the angles that u and v make with the positive x-axis. Write a program for a graphing utility in which the output is the following. (a) u + v (b) ||u + v|| (c) The angle that u + v makes with the positive x-axis (d) Use the program to find the magnitude and direction of the resultant of the vectors shown.

The initial and terminal points of vector v are (3, -4) and (9, 1), respectively. (a) Write v in component form. (b) Write v as the linear combination of the standard unit vectors i and j. (c) Sketch v with its initial point at the origin. (d) Find the magnitude of v.

In Exercises 79 and 80, use a graphing utility to find the magnitude and direction of the resultant of the vectors.

In Exercises 79 and 80, use a graphing utility to find the magnitude and direction of the resultant of the vectors.

Resultant Force Forces with magnitudes of 500 pounds and 200 pounds act on a machine part at angles of \(30^{\circ}\) and \(-45^{\circ}\), respectively, with the x-axis (see figure). Find the direction and magnitude of the resultant force. Text Transcription: 30 degrees -45 degrees

Numerical and Graphical Analysis Forces with magnitudes of 180 newtons and 275 newtons act on a hook (see figure). The angle between the two forces is \(\theta\) degrees. (a) If \(\theta=30^{\circ}\), find the direction and magnitude of the resultant force. (b) Write the magnitude M and direction \(\alpha\) of the resultant force as functions of \(\theta\), where \(0^{\circ} \leq \theta \leq 180^{\circ}\). (c) Use a graphing utility to complete the table. (d) Use a graphing utility to graph the two functions M and \(\alpha\). (e) Explain why one of the functions decreases for increasing values of \(\theta\) whereas the other does not. Text Transcription: theta theta=30 degrees alpha 0 degree leq theta leq 180 degrees

Resultant Force Three forces with magnitudes of 75 pounds, 100 pounds, and 125 pounds act on an object at angles of \(30^{\circ}, 45^{\circ}\), and \(120^{\circ}\), respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. Text Transcription: 30 degrees, 45 degree 120 degrees

Resultant Force Three forces with magnitudes of 400 newtons, 280 newtons, and 350 newtons act on an object at angles of \(-30^{\circ}, 45^{\circ}\), and \(135^{\circ}\), respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. Text Transcription: -30 degrees, 45 degrees 135 degrees

Think About It Consider two forces of equal magnitude acting on a point. (a) If the magnitude of the resultant is the sum of the magnitudes of the two forces, make a conjecture about the angle between the forces. (b) If the resultant of the forces is 0, make a conjecture about the angle between the forces. (c) Can the magnitude of the resultant be greater than the sum of the magnitudes of the two forces? Explain.

Graphical Reasoning Consider two forces \(\mathbf{F}_{1}=\langle 20,0\rangle\) and \(\mathbf{F}_{2}=10\langle\cos \theta, \sin \theta\rangle\). (a) Find \(\left\|\mathbf{F}_{1}+\mathbf{F}_{2}\right\|\). (b) Determine the magnitude of the resultant as a function of \(\theta\). Use a graphing utility to graph the function for \(0 \leq \theta<2 \pi\). (c) Use the graph in part (b) to determine the range of the function. What is its maximum and for what value of \(\theta\) does it occur? What is its minimum and for what value of \(\theta\) does it occur? (d) Explain why the magnitude of the resultant is never 0. Text Transcription: F_1=<20,0> F_2=10<cos theta, sin theta> ||F_1+F_2|| theta 0 leq theta < 2pi

Three vertices of a parallelogram are (1, 2), (3, 1), and (8, 4). Find the three possible fourth vertices (see figure).

Use vectors to find the points of trisection of the line segment with endpoints (1, 2) and (7, 5).

Cable Tension In Exercises 89 and 90, use the figure to determine the tension in each cable supporting the given load.

Cable Tension In Exercises 89 and 90, use the figure to determine the tension in each cable supporting the given load.

Projectile Motion A gun with a muzzle velocity of 1200 feet per second is fired at an angle of \(6^{\circ}\) above the horizontal. Find the vertical and horizontal components of the velocity. Text Transcription: 6 degrees

Shared Load To carry a 100-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a \(20^{\circ}\) angle away from the vertical and the other makes a \(30^{\circ}\) angle (see figure). (a) Find each rope's tension if the resultant force is vertical. (b) Find the vertical component of each worker's force. Text Transcription: 20 degrees 30 degrees

Navigation A plane is flying with a bearing of \(302^{\circ}\). Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane, and what is its speed with respect to the ground? Text Transcription: 302 degrees

Navigation A plane flies at a constant groundspeed of 400 miles per hour due east and encounters a 50-mile-per-hour wind from the northwest. Find the airspeed and compass direction that will allow the plane to maintain its groundspeed and eastward direction.

True or False? In Exercises 95-100, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If u and v have the same magnitude and direction, then u and v are equivalent.

True or False? In Exercises 95-100, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If u is a unit vector in the direction of v, then v = ||v|| u.

True or False? In Exercises 95-100, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If u = ai + bj is a unit vector, then \(a^{2}+b^{2}=1\). Text Transcription: a^2+b^2=1

True or False? In Exercises 95-100, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If v = ai + bj = 0, then a = -b.

True or False? In Exercises 95-100, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a = b, then ||ai +bj|| = \(\sqrt{2} a\). Text Transcription: sqrt 2 a

True or False? In Exercises 95-100, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If u and v have the same magnitude but opposite directions, then u + v = 0.

Prove that \(\mathbf{u}=(\cos \theta) \mathbf{i}-(\sin \theta) \mathbf{j}\) and \(\mathbf{v}=(\sin \theta) \mathbf{i}+(\cos \theta) \mathbf{j}\) are unit vectors for any angle \(\theta\). Text Transcription: u=(cos theta)i - (sin theta)j v=(sin theta)i - (cos theta)j theta

Geometry Using vectors, prove that the line segment joining the midpoints of two sides of a triangle is parallel to, and one-half the length of, the third side.

Geometry Using vectors, prove that the diagonals of a parallelogram bisect each other.

Prove that the vector w = ||u|| v + ||v|| u bisects the angle between u and v.

Consider the vector \(\mathbf{u}=\langle x, y\rangle\). Describe the set of all points (x, y) such that ||u|| = 5. Text Transcription: u=<x, y>

A coast artillery gun can fire at any angle of elevation between \(0^{\circ}\) and \(90^{\circ}\) in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant (\(=v_{0}\)). determine the set H of points in the plane and above the horizontal which can be hit. Text Transcription: 0 degree 90 degrees =v_0