Solution: In 4851, the force on an object is F = 20j . Forvector v , find:(a) The

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k a y

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k c y

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k a b

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k a z

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k c a + a y

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k a (c + y )

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k (a b )a

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k (a y )(c z )

For Exercises 19, perform the following operations on the given 3-dimensional vectors. a = 2j +  k b = 3i + 5j + 4 k c =i + 6j y = 4i 7j z =i 3j  k ((c c )a ) 

In Exercises 1014, find a normal vector to the plane. 2x + y z = 5

In Exercises 1014, find a normal vector to the plane. 2(x z) = 3(x + y)

In Exercises 1014, find a normal vector to the plane. 1.5x + 3.2y + z = 0

In Exercises 1014, find a normal vector to the plane. z = 3x + 4y 7

In Exercises 1014, find a normal vector to the plane. (x 1) = (1 )(y z) +

In Exercises 1521, find an equation of a plane that satisfies the given conditions. Through (1, 5, 2) perpendicular to 3i j + 4 k

In Exercises 1521, find an equation of a plane that satisfies the given conditions. Through (2, 1, 3) perpendicular to 5i + 4j  k .

In Exercises 1521, find an equation of a plane that satisfies the given conditions. Through (1, 3, 5) and normal toi j +  k .

In Exercises 1521, find an equation of a plane that satisfies the given conditions. Perpendicular to 5i + j 2 k and passing through (0, 1, 1).

In Exercises 1521, find an equation of a plane that satisfies the given conditions. Parallel to 2x + 4y 3z = 1 and through (1, 0, 1).

In Exercises 1521, find an equation of a plane that satisfies the given conditions. Through (2, 3, 2) and parallel to 3x + y + z = 4.

In Exercises 1521, find an equation of a plane that satisfies the given conditions. Perpendicular to v = 2i 3j + 5 k and through (4, 5, 2).

In Exercises 2226, compute the angle between the vectors. i +j +  k andi j  k .

In Exercises 2226, compute the angle between the vectors. . i +  k and j  k .

In Exercises 2226, compute the angle between the vectors. i +j  k and 2i + 3j +  k .

In Exercises 2226, compute the angle between the vectors. i +j andi + 2j  k .

In Exercises 2226, compute the angle between the vectors. i and 2i + 3j  k

Give a unit vector (a) In the same direction as v = 2i + 3j . (b) Perpendicular to v .

A plane has equation z = 5x 2y + 7. (a) Find a value of making the vector i +j + 0.5 k normal to the plane. (b) Find a value of a so that the point (a + 1, a, a 1) lies on the plane.

Consider the plane 5x y + 7z = 21. (a) Find a point on the x-axis on this plane. (b) Find two other points on the plane. (c) Find a vector perpendicular to the plane. (d) Find a vector parallel to the plane.

(a) Find a vector perpendicular to the plane z =2+3x y. (b) Find a vector parallel to the plane.

(a) Find a vector perpendicular to the plane z = 2x + 3y. (b) Find a vector parallel to the plane.

Match the planes in (a)(d) with one or more of the descriptions in (I)(IV). No reasons needed. (a) 3x y + z = 0 (b) 4x + y + 2z 5=0 (c) x + y = 5 (d) x = 5 I Goes through the origin. II Has a normal vector parallel to the xy-plane. III Goes through the point (0, 5, 0). IV Has a normal vector whose dot products withi , j ,  k are all positive.

Which pairs (if any) of vectors from the following list (a) Are perpendicular? (b) Are parallel? (c) Have an angle less than /2 between them? (d) Have an angle of more than /2 between them? a =i 3j  k , b =i +j + 2 k , c = 2i j +  k , d = i j +  k .

List any vectors that are parallel to each other and any vectors that are perpendicular to each other: v 1 =i 2j v 2 = 2i + 4j v 3 = 3i + 1.5j v 4 = 1.2i + 2.4j v 5 = 5i 2.5j v 6 = 12i 12j v 7 = 4i + 2j v 8 = 3i 6j v 9 = 0.70i 0.35j

(a) Give a vector that is parallel to, but not equal to, v = 4i + 3j . (b) Give a vector that is perpendicular to v .

For what values of t are u = t i j +  k and v = t i + tj 2 k perpendicular? Are there values of t for which u and v are parallel?

Let be the angle between v and w , with 0 < < /2. What is the effect on v w of increasing each of the following quantities? Does v w increase or decrease? (a) ||v || (b)

Write a = 3i + 2j 6 k as the sum of two vectors, one parallel, and one perpendicular, to  d = 2i 4j +  k .

Find angle BAC if A = (2, 2, 2), B = (4, 2, 1), and C = (2, 3, 1).

The points (5, 0, 0), (0, 3, 0), and (0, 0, 2) form a triangle. Find the lengths of the sides of the triangle and each of its angles.

Let S be the triangle with vertices A = (2, 2, 2), B = (4, 2, 1), and C = (2, 3, 1). (a) Find the length of the shortest side of S. (b) Find the cosine of the angle BAC at vertex A.

In Problems 4247, given v = 3i + 4j and force vector F , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . F = 4i +j

In Problems 4247, given v = 3i + 4j and force vector F , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . F = 0.2i 0.5j

In Problems 4247, given v = 3i + 4j and force vector F , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . F = 9i + 12j

In Problems 4247, given v = 3i + 4j and force vector F , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . F = 0.4i + 0.3j

In Problems 4247, given v = 3i + 4j and force vector F , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . F = 3i 5j

In Problems 4247, given v = 3i + 4j and force vector F , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . F = 6i 8

In Problems 4851, the force on an object is F = 20j . For vector v , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . v = 2i + 3j

In Problems 4851, the force on an object is F = 20j . For vector v , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . v = 5i j

In Problems 4851, the force on an object is F = 20j . For vector v , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . v = 3j

In Problems 4851, the force on an object is F = 20j . For vector v , find: (a) The component of F parallel to v . (b) The component of F perpendicular to v . (c) The work, W, done by force F through displacement v . v = 5i

A basketball gymnasium is 25 meters high, 80 meters wide and 200 meters long. For a half-time stunt, the cheerleaders want to run two strings, one from each of the two corners above one basket to the diagonally opposite corners of the gym floor. What is the cosine of the angle made by the strings as they cross?

A 100-meter dash is run on a track in the direction of the vector v = 2i + 6j . The wind velocity w is 5i + j km/hr. The rules say that a legal wind speed measured in the direction of the dash must not exceed 5 km/hr. Will the race results be disqualified due to an illegal wind? Justify your answer.

An airplane is flying toward the southeast. Which of the following wind velocity vectors increases the planes speed the most? Which slows down the plane the most? w 1 = 4i j w 2 =i 2j w 3 = i + 8j w 4 = 10i + 2j w 5 = 5i 2j

A canoe is moving with velocity v = 5i + 3j m/sec relative to the water. The velocity of the current in the water is c =i + 2j m/sec. (a) What is the speed of the current? (b) What is the speed of the current in the direction of the canoes motion?

Find a vector that bisects the smaller of the two angles formed by 3i + 4j and 5i 12j .

Find the shortest distance between the planes 2x 5y + z = 10 and z = 5y 2x.

A street vendor sells six items, with prices p1 dollars per unit, p2 dollars per unit, and so on. The vendors price vector is p = (p1, p2, p3, p4, p5, p6) = (1.00, 3.50, 4.00, 2.75, 5.00, 3.00). The vendor sells q1 units of the first item, q2 units of the second item, and so on. The vendors quantity vector is q = (q1, q2, q3, q4, q5, q6) = (43, 57, 12, 78, 20, 35). Find p q , give its units, and explain its significance to the vendor.

A course has four exams, weighted 10%, 15%, 25%, 50%, respectively. The class average on each of these exams is 75%, 91%, 84%, 87%, respectively. What do the vectors a = (0.75, 0.91, 0.84, 0.87) and w = (0.1, 0.15, 0.25, 0.5) represent, in terms of the course? Calculate the dot product w a . What does it represent, in terms of the course?

A consumption vector of three goods is defined by x = (x1, x2, x3), where x1, x2 and x3 are the quantities consumed of the three goods. A budget constraint is represented by the equation p x = k, where p is the price vector of the three goods and k is a constant. Show that the difference between two consumption vectors corresponding to points satisfying the same budget constraint is perpendicular to the price vector p .

What does Property 2 of the dot product in the box on page 735 say geometrically?

Show that the vectors (b c )a (a c )b and c are perpendicular.

Show why each of the properties of the dot product in the box on page 735 follows from the algebraic definition of the dot product: v w = v1w1 + v2w2 + v3w3.

Show that if u and v are two vectors such that u w = v w for every vector w , then u = v .

Show that u u 2 v v 2 and u u v v u v have the same magnitude where u and v are nonzero vectors.

Figure 13.34 shows that, given three vectors u , v , and w , the sum of the components of v and w in the direction of u is the component of v + w in the direction of u . (Although the figure is drawn in two dimensions, this result is also true in three dimensions.) Use this figure to explain why the geometric definition of the dot product satisfies (v + w ) u = v u + w u .

(a) Using the geometric definition of the dot product, show that u (v ) = (u v ). [Hint: What happens to the angle when you multiply v by 1?] (b) Using the geometric definition of the dot product, show that for any negative scalar u (v ) = (u v ) (u ) v = (u v ).

The Law of Cosines for a triangle with side lengths a, b, and c, and with angle C opposite side c, says c 2 = a2 + b 2 2ab cos C. On page 735, we used the Law of Cosines to show that the two definitions of the dot product are equivalent. In this problem, use the geometric definition of the dot product and its properties in the box on page 735 to prove the Law of Cosines. [Hint: Let u and v be the displacement vectors from C to the other two vertices, and express c2 in terms of u and v .]

Use Problems 66 and 67 and the following steps to show (without trigonometry) that the geometric and algebraic definitions of the dot product are equivalent. Let u = u1 i + u2j + u3 k and v = v1 i + v2j + v3 k be any vectors. Write (u v )geom for the result of the dot product computed geometrically. Substitute u = u1 i + u2j + u3 k and use Problems 6667 to expand (u v )geom. Substitute for v and expand. Then calculate the dot productsi i ,i j , etc. geometrically.

For any vectors v and w , consider the following function of t: q(t)=(v + t w ) (v + t w ). (a) Explain why q(t) 0 for all real t. (b) Expand q(t) as a quadratic polynomial in t using the properties on page 735. (c) Using the discriminant of the quadratic, show that |v w | v w .

In Problems 7173, explain what is wrong with the statement. For any 3-dimensional vectors u,v,w , we have (u v ) w = u (v w ).

In Problems 7173, explain what is wrong with the statement. If u = i +j and v = 2i +j , then the component of v parallel to u is v parallel = (v u )u = 3i + 3j .

In Problems 7173, explain what is wrong with the statement. A normal vector for the plane z = 2x + 3y is 2i + 3j .

In Problems 7475, give an example of: A point (a, b) such that the displacement vector from (1, 1) to (a, b) is perpendicular toi + 2j .

In Problems 7475, give an example of: A linear function f(x, y) = mx + ny + c whose graph is perpendicular toi + 2j + 3 k .

Are the statements in Problems 7685 true or false? Give reasons for your answer. The quantity u v is a vector.

Are the statements in Problems 7685 true or false? Give reasons for your answer. The plane x + 2y 3z = 5 has normal vectori + 2j 3 k .

Are the statements in Problems 7685 true or false? Give reasons for your answer. If u v < 0 then the angle between u and v is greater than /2

Are the statements in Problems 7685 true or false? Give reasons for your answer. An equation of the plane with normal vectori +j +  k containing the point (1, 2, 3) is z = x + y.

Are the statements in Problems 7685 true or false? Give reasons for your answer. The triangle in 3-space with vertices (1, 1, 0),(0, 1, 0) and (0, 1, 1) has a right angle.

Are the statements in Problems 7685 true or false? Give reasons for your answer. The dot product v v is never negative.

Are the statements in Problems 7685 true or false? Give reasons for your answer. If u v = 0 then either u = 0 or v = 0.

Are the statements in Problems 7685 true or false? Give reasons for your answer. If u,v and w are all nonzero, and u v = u w , then v = w .

Are the statements in Problems 7685 true or false? Give reasons for your answer. For any vectors u and v : (u + v ) (u v ) = u 2 v 2.

Are the statements in Problems 7685 true or false? Give reasons for your answer. If u = 1, then the vector v (v u )u is perpendicular to u.