Vectors Questions and Answers

A vector s has initial point (5,-2) and terminal point (-1, -3). Writes in the form s = ai + bj.

Two kickers are having a kicking competition to see who kicks farther. Bill kicks his ball with an initial velocity of 70 ft/sec at an angle of 38. Ben kicks his ball with an initial velocity of 63 ft/sec at an angle of 45°. Which kicker kicked the ball farther? Justify your answer.

At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 21 knots and ship B is sailing north at 19 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Two vessels are pulling a broken-down ship toward a boathouse with forces of 750 lb and 630 lb. The angle between the forces is 34.5°. Determine the magnitude of the equilibrant force. Show your work.

To avoid a storm, a pilot leaves airport A and flies at a bearing of 172°. He travels for one and one-half hours at a speed of 650 mph. At the end of the one and one-half hours, the pilot changes direction and flies at a bearing of 68° for two hours at the same speed. At the end of the two hours, he is at point B, and is now able to continue with a normal flight plan since he has passed the storm. If there had been no storm, the pilot would have flown directly from Airport A to point B. How much further did the plane have to travel because of the storm? b. What is the bearing of the normal flight plan (from A to B)?

Find the equation of the line and write in the specified form: a. the line parallel to d = (2, 3) that hits the point (1,4), in parametric form. b. the line that passes through the points (2, 4) and (5, 13), in vector form.

In Exercises 5-12, let P = (-2, 2), Q = (3, 4), R = (-2, 5), and S = (2, -8). Find the component form and magnitude of the vector. 9. 2QS 10. (√2) PR 11. 3QR + PS 12. PS - 3PQ

A jet leaves an airport at 3:00 pm with a course of 150° and a speed of 510 mph. At 5:00 pm, the jet pilot changes the course to 210° and keeps traveling in that direction at 510 mph. Determine how far the jet is from the airport at 8:00 pm. 310.91 miles 883.35 miles 916.56 miles 2223.04 miles

Given the points (-5, -4) and P(-3,7), determine the position vector of PQ. Select the correct answer below: (-2,11) (11,2) (-11, -2) (2,11) (-11,2) (-2,-11)

Given the points C(-3,-1) and D(-1,-6), express the vector CD in terms of i and j. Select the correct answer below: - 2i - 5j 2i - 5j -5i +2j 2i + 5j -2i+ 5j 5i-2j

Given u = (-6, -2) and 7 = (-3, 4), find the dot product u · v. Give just a number for your answer. For example, if you found u = 4, you would enter 4.

Sketch each pair of vectors below on a pair of coordinate axes and find the angle between the two vectors. Make sure to label each vector using the correct notation: a. a =< 1,- 3 > and b =< - 6,- 2 > b. c =< 2,4 > and d =< - 1,- 2 >

If two vectors are perpendicular to each other, their cross product must be zero. True False

Construct the vector 0.7 s + 1.2 t having initial point P. (Use the tools provided to move and scale the given vectors; then don't forget to draw your final answer.)

Let P = (3, 1, -2), Q = (1, t+1, 2), and R = (11, t+1,4). For which values of t will the angle between the vectors PQ and PR (a). be a right angle? If there are no such values, explain why. (b). be an acute angle (i.e. less than 90 degrees)? If there are no such values, explain why.

(a) Find the intercepts of the graph of the equation. (b) Test the equation for symmetry with respect to the x-axis, the y-axis, and the origin. (c) Graph the equation by plotting points. y=x³-9x (a) List the intercept(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The intercept(s) is/are (0,0).(3,0).(-3,0). B. There are no intercepts. (b) What are the results for the tests for symmetry? Select all that apply. A. The graph is symmetric with respect to the origin. B. The graph is symmetric with respect to the y-axis. C. The graph is symmetric with respect to the x-axis. D. The graph has no symmetry. (c) Use the graphing tool to graph the function.

A plane has normal vector n = < 1, 1, -1 > and passes through P(0, 2, -3) a. Find an equation for the plane: b. Find the intercepts of the plane. Sketch a graph of the intercepts and the plane on a set of coordinate axes.

v = 3i+j a. Sketch v on a pair of coordinate axes. b. Find the magnitude of v and its angle above or below the positive or negative x-axis. Label its magnitude and direction on the diagram above. c. Find 5v d. If u = -5j, find u- 4v

A drone is flying at 200 kilometers per hour at a constant altitude of 1 km above a straight road. The drone pilot uses radar to determine that an oncoming motorcycle is at a distance of exactly 2 kilometers from the drone, and that this distance is decreasing at 250 kph. Find the speed of the motorcycle (in kph). Hint: At the end, remember to consider that the drone is moving towards the motorcycle at 200 kph. Speed cannot be negative. Note: Round to the nearest tenth.

Consider the vectors u = < 4,5 > and v= <-2, -1 > Sketch each vector addition problem on a set of coordinate axes below, and find the components, magnitude, and direction of each resultant: a. u+v Sketch: Resultant: b. u-v Component form: Magnitude: Direction: Sketch: Resultant: Component form: Magnitude: Direction:

Which is traveling faster, a car whose velocity vector is 25i +30j, or a car whose velocity vector is 40i, assuming that the units are the same for both directions?

A bullet is fired into the air with an initial velocity of 1000 feet per second at an angle of 55° from the horizontal. The horizontal and vertical components of the velocity vector are: (Express your answers using at least three decimal places.)

A boat is heading due east at 32 km/hr (relative to the water). The current is moving toward the southwest at 11 km/hr. 1. Give the vector representing the actual movement of the boat. 2. How fast is the boat moving, relative to the ground? 3. By what angle does the current push the boat off its due east course? Your answer should be a positive angle less than π radians.

Find the x- and y-components of the given vector by use of the trigonometric functions. The magnitude is shown first, followed by the direction as an angle in standard position. 17.8 lb, 0=66.3° The x-component is (Round to two decimal places as needed.)

Determine the unit vectors in the specified directions. Write your answers in the form (₁, ₂), where U₁ and 1₂ do not involve trigonometric functions. and (a) The unit vector in the direction of (2, 5). :(. (c) The unit vector in the direction opposite (6, -2). (b) The unit vector in the direction of 26 5'5

Resolve the vector given in the indicated figure into its x-component and y-component. A=0.4-0 (Round to the nearest tenth as needed.) ***** A=726.8 0=45.0° Q Q

Find the direction angles of the given vector. Write the vector in terms of its magnitude and direction cosines as v= ||v|| [(cos α)i + (cos ß)j + (cos y)k]. v=-2i-9j+6k

Vector A points in the negative y direction and has a magnitude of 15 units. Vector B has twice the magnitude and points in the positive a direction. Find the direction and magnitude of à + B. Express your answer as a whole number.

Given the vectors in component form below. A<-3, 1> B<0, 4> C < 2, -5> D<-3, -3 > Determine B-C. <-2,9> <0, -20> not possible <2, -1>

Given vector E=<6, -1> and F=<1, 2>, find ||2E||. O 148 2√37 <12, -2> 4√37

Determine the component form for vector AB if A(-3, 4) and B(2, 0). <5, -4> <-5, 4> <-1, 4> <1, -4>

Find the unit vectors associated with the following vectors: a) <0,4> b) -3,0> c) <-3,4> d) < 2,1> e) < 5,-4>

1-4, prove that RS and PQ are equivalent by showing that they represent the same vector. 1. R = (-4, 7), S = (-1, 5), P = (0, 0), and Q = (3,-2) 2. R=(7, -3), S = (4,-5), P = (0, 0), and Q = (-3, -2) 11 3. R=(2, 1), S = (0, -1), P = (1, 4), and Q = (-1,2) 4. R = (-2, -1), S = (2, 4), P = (-3, − 1), and Q = (1, 4) - In Exercises 5-12, let P = (-2, 2), Q = (3, 4), R = (-2,5), and S = (2, -8). Find the component form and magnitude of the vector 6. RS 8. PS 5. PQ 7.OR

An airplane has an airspeed of 490 kilometers per hour bearing N45°E. The wind velocity is 40 kilometers per hour in the direction N30°W. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? Whatis its direction?

A crate is supported by two ropes. One rope makes an angle of 45° 23' with the horizontal and has a tension of 80.9 lb on it The other rope is horizontal. Find the weight of the crate and the tension in the horizontal rope.

Vectors t= -6i - 2j, u = -4i-8j, and v= -4i + 12j are give Part A: Find the angle between vectors t and u. Show all necessary calculations. (5 points) Part B: Choose a value for c, such that c> 1. Find w = cv. Show all necessary work. (2 points) Part C: Use the dot product to determine if t and w are parallel, orthogonal, or neither. Justify your answ

An object experiences two velocity vectors in its environment. V₁ = -50i + 4j V₂ = 5i + 15j What is the true speed and direction of the object? Round the speed to the thousandths place and the direction to the nearest degree. 48.847; 157° 48.847; 23° 40.792; 157° 40.792; 23°

A vector has a magnitude of 50 and a direction of 30°. Another vector has a magnitude of 60 and a direction of 150°. What are the magnitude and direction of the resultant vector? Round the magnitude to the thousandth place and the direction to the nearest degree. 55.67899° 55.678 81° 54.314: 99° 54.314; 81°

Vector v = RS has points R(-2, 12) and S(-7, 6). What are the magnitude and direction of RS? Round the answers to the thousandths place. ||v|| = 7.81; θ = 230.194° |||v|| =7.81; θ= 50.194° ||v|| =9.22; θ= 139.399° ||v|| =9.22; θ= 40.601°

Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = 3i+ 8j, v(0) = k, r(0) = i v(t) = r(t) =

A vector has initial point A (7.7, 8.8) and the terminal point B (2.7, 0.8). What are the components of the vector? x-component is 7.7 and y-component is 0.8 O x-component is -5 and y-component is -8 Ox-component is 2.7 and y-component is 8.8 O x-component is 5 and y-component is 8

Let U and W be two subspace of a finite dimensional vector space V over a Field F, Then dim U+dim W - dim(U ∩ W) Option 1: dim U+dim W - dim(U ∪ W) Option 2 dim U+dim W + dim(U ∪ W) Option 3 Option 4:none

Find the volume of the parallelepiped with adjacent edges t = 3j - 4j – 6k, u = 3i - 7j - 7k and v = 5i+j+ 3k. 94 cubic units 278 cubic units 234 cubic units 134 cubic units

This is a bonus question. You may earn up to 10 points on this question (even though the point total is listed as O). Exactly two of the following statements are true. Select the two true statements. Warning: If you pick more than two statements, you will receive zero points for this question. The dot product of two vectors is a real number. The scalar product of a vector with the number 0 is equal to the number 0. The sum of two unit vectors is a unit vector. The sum of two vectors is a vector. If a vector v is added to the vector -v, the result is the number 0. The difference of two unit vectors is a unit vector. The dot product of two vectors is equal to the angle between the two vectors. The dot product of two vectors is a vector.

Find the angle between u = = (-8, -6) and v = (12, 5). Round your answer to the nearest tenth of a degree.

If the vector u has magnitude 2 and direction 320° (relative to the positive x-axis), what is the component form (x, y) of the vector? Round your answer to the nearest hundredth. Select the correct answer below: (1.16,-1.29) (1.53,-1.29) (1.16,-0.85) (1.53,-0.85)

Consider the cone z² = x² + y² and the sphere x² + y² + z² = 18. They intersect at the point (0, 3, 3), among others. Find the angle (or the cosine of the angle) between these surfaces at the point (0,3,3).

Find the area of the parallelogram with adjacent sides u = −4i - 9j + k and v= −6i + j + 5k. about 75.3 square units about 71.5 square units about 8.7 square units about 37.7 square units

An airplane is taking off headed due north with an air speed of 173 miles per hour at an angle of 18° relative to the horizontal. The wind is blowing with a velocity of 42 miles per hour at an angle of S47°E. Find a vector that represents the resultant velocity of the plane relative to the point of takeoff. Let i point east, j point north, and k point up.

Suppose you have three vectors in component form: a = 3 ftx - 2 ftŷ + azz b = bxx + 6 ftŷ - 1 ftz c = 3 ftx + c₂y + 2 ftz and suppose you know that a + b = c. Plug the vector component expressions into this equation. Use the dot product three times to solve for the unknown vector components az, bx, cy.