For each x in (, +), let u(x) be the vector from theorigin to the point P (x, y) on the

\(3,1,-2,6,0,5\) = _____________ Equation Transcription: Text Transcription: 3, 1, -2,6, 0, 5

Suppose that u, v, and w are vectors in 3 -space such that \(\|u\|=5, u \cdot v=7\), and \(u \cdot w=-3\) (a) \(u \cdot u=\) __________ (b) \(v \cdot u=\) __________ (c) \(u \cdot(v-w)=\) ________ (d) \(u \cdot(2 w)=\) _________ Equation Transcription: Text Transcription: ||u||=5,u times v=7 uw=-3 u times u= v times u= u times (v-w)= u times (2w)=

For the vectors u and v in the preceding exercise, if the angle between u and v is \(\pi / 3\), then \(\|v\|=\)________ Equation Transcription: Text Transcription: pi/3 ||v|| =

The direction cosines of \(\langle 2,-1,3\rangle\) are \(\cos \alpha=\)_________, \(\cos \beta=\)________, and \(\cos \gamma=\)__________. Equation Transcription: ?? Text Transcription: <2,-1,3> cos alpha= cos beta= cos gamma=

The orthogonal projection of \(v=10 i\) on \(b=-3 i+j\) is __________. Equation Transcription: Text Transcription: v=10i b=-3i+j

The accompanying figure shows six vectors that are equally spaced around a circle of radius 5. Find the dot product of \(v_{0}\) with each of the other five vectors. Equation Transcription: Text Transcription: v_0

(a) Use vectors to show that \(A(2,-1,1), B(3,2,-1)\), and \(C(7,0,-2)\) are vertices of a right triangle. At which vertex is the right angle? (b) Use vectors to find the interior angles of the triangle with vertices \((-1,0),(2,-1)\), and \((1,4)\). Express your answers to the nearest degree. Equation Transcription: Text Transcription: A(2,-1,1),B(3,2,-1) C(7,0,-2) (-1,0),(2,-1) (1,4)

(a) Show that if \(v=a i+b j\) is a vector in 2 -space, then the vectors \(v_{1}=-b i+a j \text { and } v_{2}=b i-a j\) are both orthogonal to v. (b) Use the result in part (a) to find two unit vectors that are orthogonal to the vector \(v=3 i-2 j\). Sketch the vectors \(v, v_{1}\), and \(v_{2}\) Equation Transcription: Text Transcription: v=ai+bj v_1=-bi+aj and v_2=bi-aj v=3i-2j v,v_1 v_2

Explain why each of the following expressions makes no sense. (a) \(u \cdot(v \cdot w)\) (b) \((u \cdot v)+w\) (c) \(\|u \cdot v\|\) (d) \(k \cdot(u+v)\) Equation Transcription: Text Transcription: u times(v times w) (u times v)+w ||u times v|| k times (u+v)

Explain why each of the following expressions makes sense. (a) \((u \cdot v) w\) (b) \((u \cdot v)(v \cdot w)\) (c) \(u \cdot v+k\) (d) \((k u) \cdot v\) Equation Transcription: Text Transcription: (u times v)w (u times v)(v times w) u times v+k (ku)times v

Verify parts (b) and (c) of Theorem 11.3.2 for the vectors \(u=6 i-j+2 k, v=2 i+7 j+4 k, w=i+j-3 k\) and \(k=-5\) Equation Transcription: Text Transcription:

Let u = ?1, 2?, v = ?4, -2?, and w = ?6, 0?. Find (a) \(u \cdot(7 v+w)\) (b) \(\|(u \cdot w) w\|\) (c) \(\|u\|(v \cdot w)\) (d) \((\|u\| v) \cdot w\) Equation Transcription: Text Transcription: u times (7v+w) ||(u times w)w || ||u||(v times w) (||u||v)times w

Find r so that the vector from the point \(A(1,-1,3)\) to the point \(B(3,0,5)\) is orthogonal to the vector from A to the point \(P(r, r, r)\) Equation Transcription: Text Transcription: A(1,-1,3) B(3,0,5) P(r,r,r)

Find two unit vectors in 2-space that make an angle of \(45^{\circ}\) with \(4 i+3 j\) Equation Transcription: ° Text Transcription: 45 degree 4i+3j

Find the direction cosines of v and confirm that they satisfy Equation (5). Then use the direction cosines to approximate the direction angles to the nearest degree. (a) \(v=i+j-k\) (b) \(v=2 i-2 j+k\) Equation Transcription: Text Transcription: v=i+j-k v=2i-2j+k

Find the direction cosines of v and confirm that they satisfy Equation (5). Then use the direction cosines to approximate the direction angles to the nearest degree. (a) \(v=3 i-2 j-6 k\) (b) \(v=3 i-4 k\) Equation Transcription: Text Transcription: v=3i-2j-6k v=3i-4k

Show that the direction cosines of a vector satisfy \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\) Equation Transcription: Text Transcription: cos^2? alpha +cos^2? beta+cos^2 gamma? =1

Let \(\theta\) and \(\lambda\) be the angles shown in the accompanying figure. Show that the direction cosines of v can be expressed as \(\cos \alpha=\cos \lambda \cos \theta\) \(\cos \beta=\cos \lambda \sin \theta\) \(\cos \gamma=\sin \lambda\) [Hint: Express v in component form and normalize.] Equation Transcription: Text Transcription: theta lambda cos alpha ? =cos lambda cos theta cos beta = cos lambda sin theta cos gamma = sin lambda

The accompanying figure shows a cube. (a) Find the angle between the vectors \(d\) and \(u\) to the nearest degree. (b) Make a conjecture about the angle between the vectors \(d\) and \(v\), and confirm your conjecture by computing the angle. Equation Transcription: Text Transcription: d u v

Show that two nonzero vectors \(v_{1}\) and \(v_{2}\) are orthogonal if and only if their direction cosines satisfy \(\cos \alpha_{1} \cos \alpha_{2}+\cos \beta_{1}^{-} \cos \beta_{2}+\cos \gamma_{1} \cos \gamma_{2}=0\) Equation Transcription: Text Transcription: v_1 v_2 cos? alpha_1cos ?alpha_2+cos beta??_1 cos? beta_2+cos? gamma_1cos? gamma_2=0

Use the result in Exercise \(18\) to find the direction angles of the vector shown in the accompanying figure to the nearest degree. Equation Transcription: Text Transcription: 18

Find, to the nearest degree, the acute angle formed by two diagonals of a cube.

Find, to the nearest degree, the angles that a diagonal of a box with dimensions \(10 cm\) by \(15 cm\) by \(25 cm\) makes with the edges of the box. Equation Transcription: Text Transcription: 10 cm 15 cm 25 cm

In each part, find the vector component of v along b and the vector component of v orthogonal to b. Then sketch the vectors \(v, p r o j_{b} v\), and \(v-p r o j_{b} v\) (a) \(v=2 i-j, b=3 i+4 j\) (b) \(v=\langle 4,5\rangle, b=\langle 1,-2\rangle\) (c) \(v=-3 i-2 j, b=2 i+j\) Equation Transcription: ??,?? Text Transcription: ??,??

In each part, find the vector component of v along b and the vector component of v orthogonal to b. (a) \(v=2 i-j+3 k, b=i+2 j+2 k\) (b) \(v=\langle 4,-1,7\rangle, b=\langle 2,3,-6\rangle\) Equation Transcription: ??,?? Text Transcription: v=2i-j+3k,b=i+2j+2k v=?4,-1,7?,b=?2,3,-6?

Express the vector v as the sum of a vector parallel to b and a vector orthogonal to b. (a) \(v=2 i-4 j, b=i+j\) (b) \(v=3 i+j-2 k, b=2 i-k\) (c) \(v=4 i-2 j+6 k, b=-2 i+j-3 k\) Equation Transcription: Text Transcription: v=2i-4j,b=i+j v=3i+j-2k,b=2i-k v=4i-2j+6k,b=-2i+j-3k

Express the vector v as the sum of a vector parallel to b and a vector orthogonal to b. (a) \(v=\langle-3,5\rangle, b=\langle 1,1\rangle\) (b) \(v=\langle-2,1,6\rangle, b=\langle 0,-2,1\rangle\) (c) \(v=\langle 1,4,1\rangle, b=\langle 3,-2,5\rangle\) Equation Transcription: ??,?? ??,?? ??,?? Text Transcription: v=?-3,5?,b=?1,1? v=?-2,1,6?,b=?0,-2,1? v=?-1,4,1?,b=?3,-2,5?

True-False Determine whether the statement is true or false. Explain your answer. If \(a \cdot b=a \cdot c\) and \(a \neq 0\), then \(b=c\). Equation Transcription: Text Transcription: a times b=a times c a not equal to 0 b=c

True-False Determine whether the statement is true or false. Explain your answer. If v and w are nonzero orthogonal vectors, then \(v+w \neq 0\). Equation Transcription: Text Transcription: v+wn ot equal to 0

True-False Determine whether the statement is true or false. Explain your answer. If u is a unit vector that is parallel to a nonzero vector v, then \(u \cdot v=\pm\|v\|\). Equation Transcription: Text Transcription: u times v=+- ||v||

True-False Determine whether the statement is true or false. Explain your answer. If \(v\) and \(b\) are nonzero vectors, then the orthogonal projection of \(v\) on \(b\) is a vector that is parallel to \(b\). Equation Transcription: Text Transcription: v b

If L is a line in 2 -space or 3 -space that passes through the points A and B, then the distance from a point P to the line L is equal to the length of the component of the vector \(\overrightarrow{A P}\) that is orthogonal to the vector \(\overrightarrow{A B}\) (see the accompanying figure). Use this result to find the distance from the point \((P)(1,0)\) to the line through \(A(2,-3)\) and \(B(5,1)\). Equation Transcription: Text Transcription: AP^arrow AB^arrow (P)(1, 0) A(2, ?3) B(5, 1)

Use the method of Exercise 32 to find the distance from the point \(P(-3,1,2)\) to the line through \(\mathrm{A}(1,1,0)\) and \(B(-2,3,-4)\). Equation Transcription: Text Transcription: P (-3, 1, 2) A(1, 1, 0) B(-2, 3, ?4)

As shown in the accompanying figure, a child with mass 34 kg is seated on a smooth (frictionless) playground slide that is inclined at an angle of \(27^{\circ}\) with the horizontal. Estimate the force that the child exerts on the slide, and estimate how much force must be applied in the direction of P to prevent the child from sliding down the slide. Take the acceleration due to gravity to be \(9.8 \mathrm{~m} / \mathrm{s}^{2}\). Equation Transcription: 27° Text Transcription: 27 degree 9.8 m/s^2

For the child in Exercise \(34\), estimate how much force must be applied in the direction of \(Q\) (shown in the accompanying figure) to prevent the child from sliding down the slide? Equation Transcription: Text Transcription: 34 Q

Suppose that the slide in Exercise \(34\) is \(4 \ m\) long. Estimate the work done by gravity if the child slides from the top of the slide to the bottom. Equation Transcription: Text Transcription: 34 4m

A box is dragged along the floor by a rope that applies a force of \(50 \ lb\) at an angle of \(60^{\circ}\) with the floor. How much work is done in moving the box \(5 \ ft\)? Equation Transcription: 60° Text Transcription: 50 lb 60 degree 5 ft

Find the work done by a force \(F=-3 j\) pounds applied to a point that moves on a line from \((1,3)\) to \((4, 7)\). Assume that distance is measured in feet. Equation Transcription: Text Transcription: F=-3j (1, 3) (4, 7)

A force of \(F=4 i-6 j+k\) newtons is applied to a point that moves a distance of 15 meters in the direction of the vector \(i+j+k\). How much work is done? Equation Transcription: Text Transcription:

A boat travels \(100 \ meters\) due north while the wind exerts a force of \(500 \ newtons\) toward the northeast. How much work does the wind do? Equation Transcription: Text Transcription: 100 meters 500 newtons

Let \(u\) and \(v\) be adjacent sides of a parallelogram. Use vectors to prove that the diagonals of the parallelogram are perpendicular if the sides are equal in length. Equation Transcription: Text Transcription: u v

Let \(u\) and \(v\) be adjacent sides of a parallelogram. Use vectors to prove that the parallelogram is a rectangle if the diagonals are equal in length. Equation Transcription: Text Transcription: u v

Prove that \(\|u+v\|^{2}+\|u-v\|^{2}=2\|u\|^{2}+2\|v\|^{2}\) and interpret the result geometrically by translating it into a theorem about parallelograms. Equation Transcription: Text Transcription: ?u+v?^2+?u-v?^2=2?u?^2+2?v?^2

Prove: \(u+v=\frac{1}{4}\|u+v\|^{2}-\frac{1}{4}\|u-v\|^{2}\). Equation Transcription: Text Transcription: u+v=¼||u+v||^2-¼||u-v||^2

Show that if \(v_{1}, v_{2}\), and \(v_{3}\) are mutually orthogonal nonzero vectors in 3 -space, and if a vector v in 3 -space is expressed as \(v=c_{1} v_{1}+c_{2} v_{2}+c_{3} v_{3}\) then the scalars \(c_{1}, c_{2}\), and \(c_{3}\) are given by the formulas \(c_{i}=\left(v \cdot v_{i}\right) /\left\|v_{i}\right\|^{2}, i=1,2,3\) Equation Transcription: Text Transcription: v_1, v_2 v_3 v=c_1+v_1+c_2v_2+c_3+v_3 c_1, c_2 c_3 c_1=(v times v_1)/||v_1||^2, i=1,2,3

Show that the three vectors \(v_{1}=3 i-j+2 k, v_{2}=i+j-k, v_{3}=i-5 j-4 k\) are mutually orthogonal, and then use the result of Exercise 45 to find scalars \(c_{1}, c_{2}\), and \(c_{3}\) so that \(c_{1} v_{1}+c_{2} v_{2}+c_{3} v_{3}=i-j+k\) Equation Transcription: Text Transcription: v_1=3i-j+2k, v_2=i+j-k, v_3=i-5j-4k c_1,c_2 c_3 c_1v_1+c_2v_2+c_3v_3=i-j+k

For each x in \((-\infty,+\infty)\), let u(x) be the vector from the origin to the point \(P(x, y)\) on the curve \(y=x^{2}+1\), and v(x) the vector from the origin to the point \(Q(x, y)\) on the line \(y=-x-1\) (a) Use a CAS to find, to the nearest degree, the minimum angle between u(x) and v(x) for x in \((-\infty,+\infty)\). (b) Determine whether there are any real values of x for which u(x) and v(x) are orthogonal. Equation Transcription: Text Transcription: (-infinity,+infinity) P(x,y) y=x^2+1 Q(x,y) y=-x-1

Let u be a unit vector in the xy- plane of an xyz- coordinate system, and let v be a unit vector in the yz-plane. Let \(\theta_{1}\) be the angle between u and i, let \(\theta_{2}\) be the angle between v and k, and let \(\theta\) be the angle between u and v. (a) Show that \(\cos \theta=\pm \sin \theta_{1} \sin \theta_{2}\) (b) Find \(\theta\) if \(\theta\) is acute and \(\theta_{1}=\theta_{2}=45^{\circ}\). (c) Use a CAS to find, to the nearest degree, the maximum and minimum values of \(\theta\) if \(\theta\) is acute and \(\theta_{2}=2 \theta_{1}\). Equation Transcription: ° Text Transcription: theta_1 theta_2 theta cos theta=+- sin theta_1 sin theta_2 theta_1=theta_2=45 degree theta_2=2 theta_1

Prove parts \((b)\) and \((e)\) of Theorem \(11.3.2\) for vectors in 3-space. Equation Transcription: Text Transcription: (b) (e) 11.3.2

Writing Discuss some of the similarities and differences between the multiplication properties of real numbers and those of the dot product of vectors.

Writing Discuss the merits of the following claim: Suppose an algebraic identity involves only the addition, subtraction, and multiplication of real numbers. If the numbers are replaced by vectors, and the multiplication is replaced by the dot product, then an identity involving vectors will result.