Solved: Classifying a Triangle In Exercises 2528, the vertices of a triangle are given

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) \(\mathbf{u} \cdot(\mathbf{2 v})\). \(\mathbf{u}=\langle 3,4\rangle, \mathbf{v}=\langle-1,5\rangle\) Text Transcription: u dot v u dot u ||u||^2 (u dot v) v u dot(2 v) u=langle 3,4 rangle, v=langle-1,5 rangle

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) \(\mathbf{u} \cdot(\mathbf{2 v})\). \(\mathbf{u}=\langle 4,10\rangle, \mathbf{v}=\langle-2,3\rangle\) Text Transcription: u dot v u dot u ||u||^2 (u dot v) v u dot(2 v) u=langle 4,10 rangle, v=langle-2,3 rangle

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) \(\mathbf{u} \cdot(\mathbf{2 v})\). \(\mathbf{u}=\langle 6,-4\rangle, \mathbf{v}=\langle-3,2\rangle\) Text Transcription: u dot v u dot u ||u||^2 (u dot v) v u dot(2 v) u=langle 6,-4 rangle, v=langle-3,2 rangle

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) \(\mathbf{u} \cdot(\mathbf{2 v})\). \(\mathbf{u}=\langle-4,8\rangle, \mathbf{v}=\langle 7,5\rangle\) Text Transcription: u dot v u dot u ||u||^2 (u dot v) v u dot(2 v) u=langle-4,8 rangle, v=langle 7,5 rangle

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) \(\mathbf{u} \cdot(\mathbf{2 v})\). \(\mathbf{u}=\langle 2,-3,4\rangle, \mathbf{v}=\langle 0,6,5\rangle\) Text Transcription: u dot v u dot u ||u||^2 (u dot v) v u dot(2 v) u=langle 2,-3,4 rangle, v=langle 0,6,5 rangle

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 dot v u dot u ||u||^2 (u dot v) v u dot(2 v) u=i, v=i

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 dot v u dot u ||u||^2 (u dot v) v u dot(2 v)

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 dot v u dot u ||u||^2 (u dot v) v u dot(2 v)

In Exercises 9-16, find the angle \(\boldsymbol{\theta}\) between the vectors (a) in radians and (b) in degrees. \(\mathbf{u}=\langle 1,1\rangle, \mathbf{v}=\langle 2,-2\rangle\) Text Transcription: theta u=langle 1,1 rangle, v=langle 2,-2 rangle

In Exercises 9-16, find the angle \(\boldsymbol{\theta}\) between the vectors (a) in radians and (b) in degrees. \(\mathbf{u}=\langle 3,1\rangle, \mathbf{v}=\langle 2,-1\rangle\) Text Transcription: theta u=langle 3,1 rangle, v=langle 2,-1 rangle

In Exercises 9-16, find the angle \(\boldsymbol{\theta}\) between the vectors (a) in radians and (b) in degrees. u = 3i + j, v = -2i + 4j Text Transcription: theta

In Exercises 9-16, find the angle \(\boldsymbol{\theta}\) between the vectors (a) in radians and (b) in degrees. \(\mathbf{u}=\cos \left(\frac{\pi}{6}\right) \mathbf{i}+\sin \left(\frac{\pi}{6}\right) \mathbf{j}, \mathbf{v}=\cos \left(\frac{3 \pi}{4}\right) \mathbf{i}+\sin \left(\frac{3 \pi}{4}\right) \mathbf{j}\) Text Transcription: theta u=cos (frac pi 6) i+sin (frac pi 6) j,v=cos (frac3 pi 4) i+sin (frac 3 pi 4) j

In Exercises 9-16, find the angle \(\boldsymbol{\theta}\) between the vectors (a) in radians and (b) in degrees. \(\mathbf{u}=\langle 1,1,1\rangle\) \(\mathbf{v}=\langle 2,1,-1\rangle\) Text Transcription: theta u=langle 1,1,1 rangle v=langle 2,1,-1 rangle

In Exercises 9-16, find the angle \(\boldsymbol{\theta}\) between the vectors (a) in radians and (b) in degrees. u = 3i + 2j + k v = 2i - 3j Text Transcription: theta

In Exercises 9-16, find the angle \(\boldsymbol{\theta}\) between the vectors (a) in radians and (b) in degrees. u = 3i + 4j v = -2j + 3k Text Transcription: theta

In Exercises 9-16, find the angle \(\boldsymbol{\theta}\) between the vectors (a) in radians and (b) in degrees. u = 2i - 3j + k v = i - 2j + k Text Transcription: theta

In Exercises 17 and 18, use the alternative form of the dot product to find \(u \cdot v\). ||u|| = 8, ||v|| = 5, and the angle between u and v is \(\pi / 3\). Text Transcription: u dot v pi / 3

In Exercises 17 and 18, use the alternative form of the dot product to find \(u \cdot v\). ||u|| = 40, ||v|| = 25, and the angle between u and v is \(5 \pi / 6\). Text Transcription: u dot v 5 pi / 6

In Exercises 19-24, determine whether u and v are orthogonal, parallel, or neither. \(\mathbf{u}=\langle 4,3\rangle, \mathbf{v}=\left\langle\frac{1}{2},-\frac{2}{3}\right\rangle\) Text Transcription: u=langle 4,3 rangle, v=langle frac 1 2,-frac 2 3 rangle

In Exercises 19-24, determine whether u and v are orthogonal, parallel, or neither. \(\mathbf{u}=-\frac{1}{3}(\mathbf{i}-2 \mathbf{j}), \mathbf{v}=2 \mathbf{i}-4 \mathbf{j}\) Text Transcription: u=-frac 1 3 (i-2j), v=2i-4j

In Exercises 19-24, determine whether u and v are orthogonal, parallel, or neither. u = j + 6k v = i - 2j - k

In Exercises 19-24, determine whether u and v are orthogonal, parallel, or neither. u = -2i + 3j - k v = 2i + j - k

In Exercises 19-24, determine whether u and v are orthogonal, parallel, or neither. \(\mathbf{u}=\langle 2,-3,1\rangle\) \(\mathbf{v}=\langle-1,-1,-1\rangle\) Text Transcription: u=langle 2,-3,1 rangle v=langle-1,-1,-1 rangle

In Exercises 19-24, determine whether u and v are orthogonal, parallel, or neither. \(\mathbf{u}=\langle\cos \theta, \sin \theta,-1\rangle\) \(\mathbf{v}=\langle\sin \theta,-\cos \theta, 0\rangle\) Text Transcription: u=langle cos theta, sin theta,-1 rangle v=langle sin theta,-cos theta, 0 rangle

In Exercises 25-28, 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 25-28, 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 25-28, 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 25-28, 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 29-34, find the direction cosines and angles of u, and demonstrate that the sum of the squares of the direction cosines is 1 . u = i + 2j + 2k

In Exercises 29-34, find the direction cosines and angles of u, and demonstrate that the sum of the squares of the direction cosines is 1 . u = 5i + 3j - k

In Exercises 29-34, find the direction cosines and angles of u, and demonstrate that the sum of the squares of the direction cosines is 1 . u = 3i + 2j - 2k

In Exercises 29-34, find the direction cosines and angles of u, and demonstrate that the sum of the squares of the direction cosines is 1 . u = -4i + 3j + 5k

In Exercises 29-34, find the direction cosines and angles 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=langle 0,6,-4 rangle

In Exercises 29-34, find the direction cosines and angles of u, and demonstrate that the sum of the squares of the direction cosines is 1 . \(\mathbf{u}=\langle-1,5,2\rangle\) Text Transcription: u=langle-1,5,2 rangle

In Exercises 35-42, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. \(\mathbf{u}=\langle 6,7\rangle, \mathbf{v}=\langle 1,4\rangle\) Text Transcription: u=langle 6,7 rangle, v=langle 1,4 rangle

In Exercises 35-42, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. \(\mathbf{u}=\langle 9,7\rangle, \mathbf{v}=\langle 1,3\rangle\) Text Transcription: u=langle 9,7 rangle, v= langle 1,3 rangle

In Exercises 35-42, (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 35-42, (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 35-42, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. \(\mathbf{u}=\langle 0,3,3\rangle, \mathbf{v}=\langle-1,1,1\rangle\) Text Transcription: u=langle 0,3,3 rangle, v=langle-1,1,1 rangle

In Exercises 35-42, (a) find the projection of u onto v, and (b) find the vector component of u orthogonal to v. \(\mathbf{u}=\langle 8,2,0\rangle, \mathbf{v}=\langle 2,1,-1\rangle\) Text Transcription: u=langle 8,2,0 rangle, v=langle 2,1,-1 rangle

In Exercises 35-42, (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 35-42, (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.

Orthogonal Vectors State the definition of orthogonal vectors. When 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 dot (v +w) (u dot v)w u dot v + w ||u|| dot (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 when (a) the projection of u onto v equals u and (b) the projection of u onto v equals 0 ?

When 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, when (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 dot v=0 u dot v>0 u dot v<0

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 \(\mathbf{v}=\langle 2.25,2.95,2.65\rangle\) 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=langle 3240,1450,2235 rangle v=langle 2.25,2.95,2.65 rangle u dot v

Revenue Repeat Exercise 51 after increasing prices by 4%. Identify the vector operation used to increase prices by 4%.

In Exercises 53-56, 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=-frac 1 4 i+frac 3 2 j

In Exercises 53-56, find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique.) u = 9i - 4j

In Exercises 53-56, 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=langle 3,1,-2 rangle

In Exercises 53-56, 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=langle 4,-3,6 rangle

Finding an Angle Find the angle between a cubes diagonal and one of its edges.

Finding an Angle Find the angle between the diagonal of a cube and the diagonal of one of its sides.

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 (b) the force perpendicular to the hill. Text Transcription: 10^circ

A 5400-pound sport utility vehicle is parked on an \(18^{\circ}\) slope. Assume the only force to overcome is that due to gravity. Find (a) the force required to keep the vehicle from rolling down the hill and (b) the force perpendicular to the hill. Text Transcription: 18^circ

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^circ

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). Find the work done in pulling the wagon 50 feet. Text Transcription: 20^circ

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^circ

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

In Exercises 65 and 66, 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 dot v = u dot w u neq 0

In Exercises 65 and 66, 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.

In Exercises 67-70, (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^circ leq theta leq 90^circ) y=x^2, y=x^1 / 3

In Exercises 67-70, (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^circ leq theta leq 90^circ) y=x^3, y=x^1 / 3

In Exercises 67-70, (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^circ leq theta leq 90^circ) y=1-x^2, y=x^2-1

In Exercises 67-70, (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^circ leq theta leq 90^circ) (y+1)^2=x, y=x^3-1

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

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

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 \(\mathrm{CH}_{4}\) or \(\mathrm{PbCl}_{4}\), where the structure of the molecule is a tetrahedron. Text Transcription: CH_4 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=langle cos alpha, sin alpha, 0 rangle v=langle cos beta, sin beta, 0 rangle 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 dot v

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

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

Proof Prove Theorem 11.6.