Get answer: True or False In Exercises 77 and 78, determine whether the statement is

In Exercises 1-8, find (a) \(\mathbf{u} \cdot \mathbf{v}\), (b) \(\mathbf{u} \cdot \mathbf{u}\), (c) \(\|\mathbf{u}\|^{2}\), (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\), and (e) \(\mathbf{u} \cdot(\mathbf{2 v})\), \(\mathbf{u}=\langle 3,4\rangle, \quad \mathbf{v}=\langle-1,5\rangle\) Text Transcription: u times v u times u ||u||^2 (u times v)v u times (2v) u=<3, 4>, v=<-1, 5>

In Exercises 1-8, find (a) \(\mathbf{u} \cdot \mathbf{v}\), (b) \(\mathbf{u} \cdot \mathbf{u}\), (c) \(\|\mathbf{u}\|^{2}\), (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\), and (e) \(\mathbf{u} \cdot(\mathbf{2 v})\), \(\mathbf{u}=\langle 4,10\rangle, \quad \mathbf{v}=\langle-2,3\rangle\) Text Transcription: u times v u times u ||u||^2 (u times v)v u times (2v) u=<4, 10>, v=<-2, 3>

In Exercises 1-8, find (a) \(\mathbf{u} \cdot \mathbf{v}\), (b) \(\mathbf{u} \cdot \mathbf{u}\), (c) \(\|\mathbf{u}\|^{2}\), (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\), and (e) \(\mathbf{u} \cdot(\mathbf{2 v})\), \(\mathbf{u}=\langle 6,-4\rangle, \quad \mathbf{v}=\langle-3,2\rangle\) Text Transcription: u times v u times u ||u||^2 (u times v)v u times (2v) u=<6, -4>, v=<-3, 2>

In Exercises 1-8, find (a) \(\mathbf{u} \cdot \mathbf{v}\), (b) \(\mathbf{u} \cdot \mathbf{u}\), (c) \(\|\mathbf{u}\|^{2}\), (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\), and (e) \(\mathbf{u} \cdot(\mathbf{2 v})\), \(\mathbf{u}=\langle-4,8\rangle, \quad \mathbf{v}=\langle 7,5\rangle\) Text Transcription: u times v u times u ||u||^2 (u times v)v u times (2v) u=<-4, 8>, v=<7, 5>

In Exercises 1-8, find (a) \(\mathbf{u} \cdot \mathbf{v}\), (b) \(\mathbf{u} \cdot \mathbf{u}\), (c) \(\|\mathbf{u}\|^{2}\), (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\), and (e) \(\mathbf{u} \cdot(\mathbf{2 v})\), \(\mathbf{u}=\langle 2,-3,4\rangle, \quad \mathbf{v}=\langle 0,6,5\rangle\) Text Transcription: u times v u times u ||u||^2 (u times v)v u times (2v) u=<2, -3, 4>, v=<0, 6, 5>

In Exercises 1-8, find (a) \(\mathbf{u} \cdot \mathbf{v}\), (b) \(\mathbf{u} \cdot \mathbf{u}\), (c) \(\|\mathbf{u}\|^{2}\), (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\), and (e) \(\mathbf{u} \cdot(\mathbf{2 v})\), u = i, v = i Text Transcription: u times v u times u ||u||^2 (u times v)v u times (2v)

In Exercises 1-8, find (a) \(\mathbf{u} \cdot \mathbf{v}\), (b) \(\mathbf{u} \cdot \mathbf{u}\), (c) \(\|\mathbf{u}\|^{2}\), (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\), and (e) \(\mathbf{u} \cdot(\mathbf{2 v})\), u = 2i - j + k v = i - k Text Transcription: u times v u times u ||u||^2 (u times v)v u times (2v)

In Exercises 1-8, find (a) \(\mathbf{u} \cdot \mathbf{v}\), (b) \(\mathbf{u} \cdot \mathbf{u}\), (c) \(\|\mathbf{u}\|^{2}\), (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\), and (e) \(\mathbf{u} \cdot(\mathbf{2 v})\), u = 2i + j - 2k v = i - 3j + 2k Text Transcription: u times v u times u ||u||^2 (u times v)v u times (2v)

In Exercises 9 and 10, find \(\mathbf{u} \cdot \mathbf{v}\). ||u|| = 8, ||v|| = 5, and the angle between u and v is \(\pi\)/3. Text Transcription: u times v pi

In Exercises 9 and 10, find \(\mathbf{u} \cdot \mathbf{v}\). ||u|| = 40, ||v|| = 25, and the angle between u and v is 5\(\pi\)/6. Text Transcription: u times v pi

In Exercises 11-18, find the angle \(\theta\) between the vectors. \(\mathbf{u}=\langle 1,1\rangle, \mathbf{v}=\langle 2,-2\rangle\) Text Transcription: theta u = <1, 1>, v = <2, -2>

In Exercises 11-18, find the angle \(\theta\) between the vectors. \(\mathbf{u}=\langle 3,1\rangle, \mathbf{v}=\langle 2,-1\rangle\) Text Transcription: theta u = <3, 1>, v = <2, -1>

In Exercises 11-18, find the angle \(\theta\) between the vectors. u = 3i + j, v = -2i + 4j Text Transcription: theta

In Exercises 11-18, find the angle \(\theta\) between the vectors. \(\begin{array}{l} \mathbf{u}=\cos \left(\frac{\pi}{6}\right) \mathbf{i}+\sin \left(\frac{\pi}{6}\right) \mathbf{j} \\ \mathbf{v}=\cos \left(\frac{3 \pi}{4}\right) \mathbf{i}+\sin \left(\frac{3 \pi}{4}\right) \mathbf{j} \end{array} \) Text Transcription: theta u=cos(pi/6)i + sin(pi/6)j v=cos(3pi/4)i + sin(3pi/4)j

In Exercises 11-18, find the angle \(\theta\) between the vectors. \(\begin{array}{l}\mathbf{u}=\langle 1,1,1\rangle \\\mathbf{v}=\langle 2,1,-1\rangle\end{array}\) Text Transcription: theta u = <1, 1, 1> v = <2, 1, -1>

In Exercises 11-18, find the angle \(\theta\) between the vectors. u = 3i + 2j + k v = 2i - 3j Text Transcription: theta

In Exercises 11-18, find the angle \(\theta\) between the vectors. u = 3i + 4j v = -2j + 3k Text Transcription: theta

In Exercises 11-18, find the angle \(\theta\) between the vectors. u = 2i - 3j + k v = i - 2j+ k Text Transcription: theta

In Exercises 19-26, determine whether u and v are orthogonal, parallel, or neither. \(\mathbf{u}=\langle 4,0\rangle, \quad \mathbf{v}=\langle 1,1\rangle\) Text Transcription: u=<4, 0>, v=<1, 1>

In Exercises 19-26, determine whether u and v are orthogonal, parallel, or neither. \(\mathbf{u}=\langle 2,18\rangle, \quad \mathbf{v}=\left\langle\frac{3}{2},-\frac{1}{6}\right\rangle\) Text Transcription: u=<2, 18>, v=<3/2, -1/6>

In Exercises 19-26, determine whether u and v are orthogonal, parallel, or neither. \(\begin{array}{l}\mathbf{u}=\langle 4,3\rangle \\\mathbf{v}=\left\langle\frac{1}{2},-\frac{2}{3}\right\rangle\end{array}\) Text Transcription: u=<4, 3> v=<1/2, -2/3>

In Exercises 19-26, determine whether u and v are orthogonal, parallel, or neither. \(\begin{array}{l}\mathbf{u}=-\frac{1}{3}(\mathbf{i}-2 \mathbf{j}) \\\mathbf{v}=2 \mathbf{i}-4 \mathbf{j}\end{array}\) Text Transcription: u=-1/3(i-2j) v=2i-4j

In Exercises 19-26, determine whether u and v are orthogonal, parallel, or neither. u = j + 6k v = i - 2j - k

In Exercises 19-26, determine whether u and v are orthogonal, parallel, or neither. u = -2i + 3j - k v = 2i + j - k

In Exercises 19-26, determine whether u and v are orthogonal, parallel, or neither. \(\begin{array}{l}\mathbf{u}=\langle 2,-3,1\rangle \\\mathbf{v}=\langle-1,-1,-1\rangle\end{array}\) Text Transcription: u=<2, -3, 1> v=<-1, -1, -1>

In Exercises 19-26, determine whether u and v are orthogonal, parallel, or neither. \(\begin{array}{l}\mathbf{u}=\langle\cos \theta, \sin \theta,-1\rangle \\\mathbf{v}=\langle\sin \theta,-\cos \theta, 0\rangle\end{array}\) Text Transcription: u=<cos theta, sin theta, -1> v=<sin theta, -cos theta, 0>

In Exercises 27-30, the vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. (1, 2, 0), (0, 0, 0), (-2, 1,0)

In Exercises 27-30, the vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. (-3, 0, 0), (0, 0, 0), (1, 2, 3)

In Exercises 27-30, the vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. (2, 0, 1), (0, 1, 2), (-0.5, 1.5, 0)

In Exercises 27-30, the vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. (2, -7, 3), (-1, 5, 8), (4, 6, -1)

In Exercises 31-34, find the direction cosines of u and demonstrate that the sum of the squares of the direction cosines is 1. u = i + 2j + 2k

In Exercises 31-34, find the direction cosines of u and demonstrate that the sum of the squares of the direction cosines is 1. u = 5i + 3j - k

In Exercises 31-34, find the direction cosines of u and demonstrate that the sum of the squares of the direction cosines is 1. \(\mathbf{u}=\langle 0,6,-4\rangle\) Text Transcription: u=<0,6,-4>

In Exercises 31-34, find the direction cosines of u and demonstrate that the sum of the squares of the direction cosines is 1. \(\mathbf{u}=\langle a, b, c\rangle\) Text Transcription: u=<a,b,c>

In Exercises 35-38, find the direction angles of the vector. u = 3i + 2j - 2k

In Exercises 35-38, find the direction angles of the vector. u = -4i + 3j + 5k

In Exercises 35-38, find the direction angles of the vector. \(\mathbf{u}=\langle-1,5,2\rangle\) Text Transcription: u=<-1, 5, 2>

In Exercises 35-38, find the direction angles of the vector. \(\mathbf{u}=\langle-2,6,1\rangle\) Text Transcription: u=<-2, 6, 1>

In Exercises 39 and 40, use a graphing utility to find the magnitude and direction angles of the resultant of forces \(F_{1}\) and \(F_{2}\) with initial points at the origin. The magnitude and terminal point of each vector are given. Vector Magnitude Terminal Point \(F_{1}\) 50 lb (10,5,3) \(F_{2}\) 80 lb (12,7,-5) Text Transcription: F_1 F_2

In Exercises 39 and 40, use a graphing utility to find the magnitude and direction angles of the resultant of forces \(F_{1}\) and \(F_{2}\) with initial points at the origin. The magnitude and terminal point of each vector are given. Vector Magnitude Terminal Point \(F_{1}\) 300 N (-20,-10,5) \(F_{2}\) 100 N (5,15,0) Text Transcription: F_1 F_2

Load-Supporting Cables A load is supported by three cables, as shown in the figure. Find the direction angles of the load-supporting cable OA.

Load-Supporting Cables The tension in the cable OA in Exercise 41 is 200 newtons. Determine the weight of the load.

In Exercises 43-50, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. \(\mathbf{u}=\langle 6,7\rangle, \quad \mathbf{v}=\langle 1,4\rangle\) Text Transcription: u=<6,7>, v=<1,4>

In Exercises 43-50, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. \(\mathbf{u}=\langle 9,7\rangle, \quad \mathbf{v}=\langle 1,3\rangle\) Text Transcription: u=<9,7>, v=<1,3>

In Exercises 43-50, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. u = 2i + 3j, v = 5i + j

In Exercises 43-50, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. u = 2i - 3j. v = 3i + 2j

In Exercises 43-50, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. \(\mathbf{u}=\langle 0,3,3\rangle, \quad \mathbf{v}=\langle-1,1,1\rangle\) Text Transcription: u=<0,3,3>, v=<-1,1,1>

In Exercises 43-50, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. \(\mathbf{u}=\langle 8,2,0\rangle, \quad \mathbf{v}=\langle 2,1,-1\rangle\) Text Transcription: u=<8,2,0>, v=<2,1,-1>

In Exercises 43-50, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. u = 2i + j + 2k, v = 3j + 4k

In Exercises 43-50, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. u = i + 4k, v = 3i + 2k

Define the dot product of vectors u and v.

State the definition of orthogonal vectors. If vectors are neither parallel nor orthogonal, how do you find the angle between them? Explain.

Determine which of the following are defined for nonzero vectors u, v, and w. Explain your reasoning. (a) \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})\) (b) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\) (c) \(\mathbf{u} \cdot \mathbf{v}+\mathbf{w}\) (d) \(\|\mathbf{u}\| \cdot(\mathbf{v}+\mathbf{w})\) Text Transcription: u times (v+w) (u times v)w u times v+w ||u|| times (v+w)

Describe direction cosines and direction angles of a vector v.

Give a geometric description of the projection of u onto v.

What can be said about the vectors u and v if (a) the projection of u onto v equals u and (b) the projection of u onto v equals 0?

If the projection of u onto v has the same magnitude as the projection of v onto u, can you conclude that ||u|| = ||v||? Explain.

What is known about \(\theta\), the angle between two nonzero vectors u and v, if (a) \(\mathbf{u} \cdot \mathbf{v}=0\)? (b) \(\mathbf{u} \cdot \mathbf{v}>0\)? (c) \(\mathbf{u} \cdot \mathbf{v}<0\)? Text Transcription: theta u times v = 0 u times v > 0 u times v < 0

Revenue The vector \(\mathbf{u}=\langle 3240,1450,2235\rangle\) gives the numbers of hamburgers, chicken sandwiches, and cheeseburgers, respectively, sold at a fast-food restaurant in one week. The vector \(\mathbf{v}=\langle 1.35,2.65,1.85\rangle\) gives the prices (in dollars) per unit for the three food items. Find the dot product \(\mathbf{u} \cdot \mathbf{v}\), and explain what information it gives. Text Transcription: u = <3240, 1450, 2235> v = <1.35, 2.65, 1.85> u cdot v

Revenue Repeat Exercise 59 after increasing prices by 4%. Identify the vector operation used to increase prices by 4%.

Programming Given vectors u and v in component form, write a program for a graphing utility in which the output is (a) ||u|| (b) ||v||, and (c) the angle between u and v.

Programming Use the program you wrote in Exercise 61 to find the angle between the vectors \(\mathbf{u}=\langle 8,-4,2\rangle\) and \(\mathbf{v}=\langle 2,5,2\rangle\). Text Transcription: u=<8,-4,2> v=<2,5,2>

Programming Given vectors u and v in component form, write a program for a graphing utility in which the output is the component form of the projection of u onto v.

Programming Use the program you wrote in Exercise 63 to find the projection of u onto v for \(\mathbf{u}=\langle 5,6,2\rangle\) and \(\mathbf{v}=\langle-1,3,4\rangle\). Text Transcription: u=<5,6,2> v=<-1,3,4>

Think About It In Exercises 65 and 66, use the figure to determine mentally the projection of u onto v. (The coordinates of the terminal points of the vectors in standard position are given.) Verify your results analytically.

Think About It In Exercises 65 and 66, use the figure to determine mentally the projection of u onto v. (The coordinates of the terminal points of the vectors in standard position are given.) Verify your results analytically.

In Exercises 67-70, find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique.) \(\mathbf{u}=-\frac{1}{4} \mathbf{i}+\frac{3}{2} \mathbf{j}\) Text Transcription: u=-1/4i+3/2j

In Exercises 67-70, find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique.) \(\mathbf{u}=9 \mathbf{i}-4 \mathbf{j}\) Text Transcription: u=9i-4j

In Exercises 67-70, find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique.) \(\mathbf{u}=\langle 3,1,-2\rangle\) Text Transcription: u=<3,1,-2>

In Exercises 67-70, find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique.) \(\mathbf{u}=\langle 4,-3,6\rangle\) Text Transcription: u=<4,-3,6>

Braking Load A 48,000-pound truck is parked on a \(10^{\circ}\) slope (see figure). Assume the only force to overcome is that due to gravity. Find (a) the force required to keep the truck from rolling down the hill and (6) the force perpendicular to the hill. Text Transcription: 10 degrees

Load-Supporting Cables Find the magnitude of the projection of the load-supporting cable onto the positive z-axis as shown in the figure.

Work An object is pulled 10 feet across a floor, using a force of 85 pounds. The direction of the force is \(60^{\circ}\) above the horizontal (see figure). Find the work done. Text Transcription: 60 degrees

Work A toy wagon is pulled by exerting a force of 25 pounds on a handle that makes a \(20^{\circ}\) angle with the horizontal (see figure in left column). Find the work done in pulling the wagon 50 feet. Text Transcription: 20 degrees

Work A car is towed using a force of 1600 newtons. The chain used to pull the car makes a \(25^{\circ}\) angle with the horizontal. Find the work done in towing the car 2 kilometers. Text Transcription: 25 degrees

Work A sled is pulled by exerting a force of 100 newt a rope that makes a \(25^{\circ}\) angle with the horizontal. Find the work done in pulling the sled 40 meters. Text Transcription: 25 degrees

True or False? In Exercises 77 and 78, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{u} \cdot \mathbf{v}=\mathbf{u} \cdot \mathbf{w}\) and \(\mathbf{u} \neq \mathbf{0}\), then v = w. Text Transcription: u times v = u times w u neq 0

True or False? In Exercises 77 and 78, 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 are orthogonal to w, then u + v is orthogonal to w.

True or False? In Exercises 77 and 78, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Find the angle between a cube's diagonal and one of its edges.

True or False? In Exercises 77 and 78, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Find the angle between the diagonal of a cube and the diagonal of one of its sides.

In Exercises 81-84, (a) find all points of intersection of the graphs of the two equations, (b) find the unit tangent vectors to each curve at their points of intersection, and (c) find the angles \(\left(0^{\circ} \leq \theta \leq 90^{\circ}\right)\) between the curves at their points of intersection. \(y=x^{2}, \quad y=x^{1 / 3}\) Text Transcription: (0 degrees leq theta leq 90 degrees) y=x^2, y=x^1/3

In Exercises 81-84, (a) find all points of intersection of the graphs of the two equations, (b) find the unit tangent vectors to each curve at their points of intersection, and (c) find the angles \(\left(0^{\circ} \leq \theta \leq 90^{\circ}\right)\) between the curves at their points of intersection. \(y=x^{3}, \quad y=x^{1 / 3}\) Text Transcription: (0 degrees leq theta leq 90 degrees) y=x^3, y=x^1/3

In Exercises 81-84, (a) find all points of intersection of the graphs of the two equations, (b) find the unit tangent vectors to each curve at their points of intersection, and (c) find the angles \(\left(0^{\circ} \leq \theta \leq 90^{\circ}\right)\) between the curves at their points of intersection. \(y=1-x^{2}, \quad y=x^{2}-1\) Text Transcription: (0 degrees leq theta leq 90 degrees) y=1-x^2, y=x^2-1

In Exercises 81-84, (a) find all points of intersection of the graphs of the two equations, (b) find the unit tangent vectors to each curve at their points of intersection, and (c) find the angles \(\left(0^{\circ} \leq \theta \leq 90^{\circ}\right)\) between the curves at their points of intersection. \((y+1)^{2}=x, \quad y=x^{3}-1\) Text Transcription: (0 degrees leq theta leq 90 degrees) (y+1)^2=x, y=x^3-1

Use vectors to prove that the diagonals of a rhombus are perpendicular.

Use vectors to prove that a parallelogram is a rectangle if and only if its diagonals are equal in length.

Bond Angle Consider a regular tetrahedron with vertices (0, 0,0), (k, k, 0), (k, 0, k), and (0, k, k), where k is a positive real number. (a) Sketch the graph of the tetrahedron. (b) Find the length of each edge. (c) Find the angle between any two edges. (d) Find the angle between the line segments from the centroid (k/2, k/2, k/2) to two vertices. This is the bond angle for a molecule such as \(\mathrm{CH}_{4} \text { or } \mathrm{PbCl}_{4}\), where the structure of the molecule is a tetrahedron. Text Transcription: CH_4 or PbCl_4

Consider the vectors \(\mathbf{u}=\langle\cos \alpha, \sin \alpha, 0\rangle\) and \(\mathbf{v}=\langle\cos \beta, \sin \beta, 0\rangle\), where \(alpha>\beta\). Find the dot product of the vectors and use the result to prove the identity \(\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\). Text Transcription: u=<cos alpha, sin alpha, 0> v=<cos beta, sin beta, 0> alpha < beta cos(alpha-beta)=cos alpha cos beta + sin alpha sin beta

Prove that \(\|\mathbf{u}-\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}-2 \mathbf{u} \cdot \mathbf{v}\). Text Transcription: ||u-v||^2=||u||^2+||v||^2-2u times v

Prove the Cauchy-Schwarz Inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq\|\mathbf{u}\|\|\mathbf{v}\|\). Text Transcription: |u times v| leq ||u|| ||v||

Prove the triangle inequality \(\|\mathbf{u}+\mathbf{v}\| \leq\|\mathbf{u}\|+\|\mathbf{v}\|\). Text Transcription: ||u+v|| leq ||u|| + ||v||

Prove Theorem 11.6.