Solved: Angles Between Lines in the PlaneThe acute angle

Problem 1E Dot Product and Projections In Exercise, find a. v . u, b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector projv u. V = 2i u =

Problem 2E Dot Product and Projections In Exercise, find a. v . u, b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector projv u. V = u =

Problem 3E Dot Product and Projections In Exercise, find a. v . u, b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector projv u. v = 10i + 11j - 2k, u = 3j + 4k

Problem 4E Dot Product and Projections In Exercise, find a. v . u b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector projv u. v = 2i + 10j - 11k, u = 2i + 2j + k

Problem 5E Dot Product and Projections In Exercise, find a. v . u b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector projv u. v = 5j - 3k, u = i + j + k

Problem 6E Dot Product and Projections In Exercise, find a. v . u b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector projv u.

Problem 7E Dot Product and Projections In Exercise, find a. v . u b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector projv u.

Problem 8E Dot Product and Projections In Exercise, find a. v . u b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector projv u.

Problem 9E Angle Between Vectors Find the angles between the vectors to the nearest hundredth of a radian. u = 2i + j, v = i + 2j – k

Problem 10E Angle Between Vectors Find the angles between the vectors to the nearest hundredth of a radian. u = 2i - 2j + k, v = 3i + 4k

Problem 11E Angle Between Vectors Find the angles between the vectors to the nearest hundredth of a radian. u = v =

Problem 12E Angle Between Vectors Find the angles between the vectors to the nearest hundredth of a radian. u = v =

Problem 13E Angle Between Vectors Triangle Find the measures of the angles of the triangle whose vertices are A = ( -1, 0), B = (2, 1) , and C = (1, -2) .

Problem 14E Angle Between Vectors Rectangle Find the measures of the angles between the diagonals of the rectangle whose vertices are A = (1, 0), B = (0, 3), C = (3, 4) , and D = (4, 1) .

Problem 15E Angle Between Vectors Direction angles and direction cosines The direction angles ?, ?, and ? of a vector v = ai + bj + ck are defined as follows: ? is the angle between v and the positive x-axis . ? is the angle between v and the positive y-axis . ? is the angle between v and the positive z-axis a. Show that and 222=1. These cosines are called the direction cosines of v. b. Unit vectors are built from direction cosines Show that if v = ai + bj + ck is a unit vector, then a, b, and c are the direction cosines of v.

Problem 16E Angle Between Vectors Water main construction A water main is to be constructed with a 20% grade in the north direction and a 10% grade in the east direction. Determine the angle ? required in the water main for the turn from north to east.

Problem 17E Theory and Examples Sums and differences In the accompanying figure, it looks as if v1 + v2 and v1 - v2 are orthogonal. Is this mere coincidence, or are there circumstances under which we may expect the sum of two vectors to be orthogonal to their difference? Give reasons for your answer.

Problem 18E Theory and Examples Orthogonality on a circle Suppose that AB is the diameter of a circle with center O and that C is a point on one of the two arcs joining A and B. Show that are orthogonal.

Problem 19E Theory and Examples Diagonals of a rhombus Show that the diagonals of a rhombus (parallelogram with sides of equal length) are perpendicular.

Problem 20E Theory and Examples Perpendicular diagonals Show that squares are the only rectangles with perpendicular diagonals.

Problem 21E Theory and Examples When parallelograms are rectangles Prove that a parallelogram is a rectangle if and only if its diagonals are equal in length. (This fact is often exploited by carpenters.)

Problem 22E Theory and Examples Diagonal of parallelogram Show that the indicated diagonal of the parallelogram determined by vectors u and v bisects the angle between u and v if

Problem 23E Theory and Examples Projectile motion A gun with muzzle velocity of 1200 ft /sec is fired at an angle of 8° above the horizontal. Find the horizontal and vertical components of the velocity.

Problem 24E Theory and Examples Inclined plane Suppose that a box is being towed up an inclined plane as shown in the figure. Find the force w needed to make the component of the force parallel to the inclined plane equal to 2.5 lb.

Problem 25E Theory and Examples a. Cauchy-Schwartz inequality Since u. v = , show that the inequality holds for any vectors u and v. b. Under what circumstances, if any, does Give reasons for your answer.

Problem 26E Dot multiplication is positive definite Show that dot multiplication of vectors is positive definite; that is, show that u . u 0 for every vector u and that u . u = 0 if and only if u = 0

Problem 27E Theory and Examples Orthogonal unit vectors If and are orthogonal unit vectors and v = u1 + u2, find v . u1.

Problem 28E Theory and Examples Cancellation in dot products In real-number multiplication, if 1 = 2 and we can cancel the u and conclude that 1 = 2. Does the same rule hold for the dot product? That is, if u . v1 = u . v2 and u can you conclude that v1 = v2 ? Give reasons for your answer.

Problem 29E Theory and Examples Using the definition of the projection of u onto v, show by direct calculation that ( u projv u) . projv u = 0

Problem 30E Theory and Examples A force F = 2i + j - 3k is applied to a spacecraft with velocity vector v = 3i - j. Express F as a sum of a vector parallel to v and a vector orthogonal to v.

Problem 31E Equations for Lines in the Plane Line perpendicular to a vector Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope of the vector v is the negative reciprocal of the slope of the given line.

Problem 32E Equations for Lines in the Plane Line parallel to a vector Show that the vector v = ai + bj is parallel to the line bx - ay = c by establishing that the slope of the line segment representing v is the same as the slope of the given line.

Problem 33E Equations for Lines in the Plane In Exercise, use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin. P (2, 1), v = i + 2j

Problem 34E Equations for Lines in the Plane In Exercise, use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin. P(-1,2), v = -2i - j

Problem 35E Equations for Lines in the Plane In Exercise, use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin. P(-2, -7), v = -2i + j

Problem 36E Equations for Lines in the Plane In Exercise, use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin. P(11, 10), v = 2i - 3j

Problem 37E Equations for Lines in the Plane In Exercises, use the result of Exercise 32 to find an equation for the line through P parallel to v. Then sketch the line. Include v in your sketch as a vector starting at the origin.. P(-2,1), v = i - j.

Problem 38E Equations for Lines in the Plane In Exercises, use the result of Exercise 32 to find an equation for the line through P parallel to v. Then sketch the line. Include v in your sketch as a vector starting at the origin. P(0, -2), v = 2i + 3j

Problem 39E Equations for Lines in the Plane In Exercises, use the result of Exercise 32 to find an equation for the line through P parallel to v. Then sketch the line. Include v in your sketch as a vector starting at the origin. P(1, 2), v = -i -2j.

Problem 40E Equations for Lines in the Plane In Exercises, use the result of Exercise 32 to find an equation for the line through P parallel to v. Then sketch the line. Include v in your sketch as a vector starting at the origin. P(1,3), v = 3i - 2j

Problem 41E Work Work along a line Find the work done by a force F = 5i (magnitude 5 N) in moving an object along the line from the origin to the point (1, 1) (distance in meters).

Problem 42E Work Locomotive The Union Pacific’s Big Boy locomotive could pull 6000-ton trains with a tractive effort (pull) of 602,148 N (135,375 lb). At this level of effort, about how much work did Big Boy do on the (approximately straight) 605-km journey from San Francisco to Los Angeles?

Problem 43E Work Inclined plane How much work does it take to slide a crate 20 m along a loading dock by pulling on it with a 200 N force at an angle of 30° from the horizontal?

Problem 44E Work Sailboat The wind passing over a boat’s sail exerted a 1000-lb magnitude force F as shown here. How much work did the wind perform in moving the boat forward 1 mi? Answer in foot-pounds.

Problem 45E Angles Between Lines in the Plane The acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines. Use this fact and the results of Exercise 31 or 32 to find the acute angles between the lines in Exercise 3 2

Problem 46E Angles Between Lines in the Plane The acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines. Use this fact and the results of Exercise 31 or 32 to find the acute angles between the lines in Exercise

Problem 47E Angles Between Lines in the Plane The acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines. Use this fact and the results of Exercise 31 or 32 to find the acute angles between the lines in Exercise

Problem 48E Angles Between Lines in the Plane The acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines. Use this fact and the results of Exercise 31 or 32 to find the acute angles between the lines in Exercise (1))

Problem 49E Angles Between Lines in the Plane The acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines. Use this fact and the results of Exercise 31 or 32 to find the acute angles between the lines in Exercise 3x - 4y = 3, x - y = 7

Problem 50E Angles Between Lines in the Plane The acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines. Use this fact and the results of Exercise 31 or 32 to find the acute angles between the lines in Exercise 12x + 5y = 1, 2x - 2y = 3