Deck 13: A Fundamental Tool- Vectors

Question 1

An airplane wants to travel east, but there is a wind of 100 miles per hour blowing from the northwest.If the airspeed of the plane is 390 miles per hour, find the direction the plane should head (as an angle north of west)and its groundspeed.

Answer 1

312.83 miles per hour; heading W N.

Question 2

Find a vector of length 5 that points in the same direction as the displacement vector from the point P(-1, 2, 2)to the point Q(3, 6, 5).

Answer 2

Question 3

If and , find the values and such that .

Answer 3

Question 4

True or false: The vector is a unit vector provided .Give a reason for your answer.

Question 5

If the vector has magnitude 9, find a.

Question 6

The vector is the displacement vector from the point (1, -1, 3)to which other point?

Question 7

Find a unit vector that points in the same direction as the displacement vector from the point P(0, 2, -3 )to the point Q(3, 5, 5).

Question 8

Three men are trying to hold a ferocious lion still for the veterinarian.The lion, in the center, is wearing a collar with three ropes attached to it and each man has hold of a rope.Charlie is pulling in the direction N 60° W with a force of 380 pounds and Sam is pulling in the direction N 44° E with a force of 400 pounds.
What is the direction and magnitude of the force needed on the third rope to counterbalance Sam and Charlie?

Question 9

If the vector is parallel to find a and b.

Question 10

A bird is flying with velocity (measured in m/ sec )relative to the air.The wind is blowing at a speed of 4 m/ sec parallel to the x-axis but opposing the bird's motion.Find the speed of the bird relative to the ground.

Question 11

Let $\vec { v } = 3 \vec { i } + 2 \vec { j } - 3 \vec { k }$ .Which of the following are parallel to $\vec { v } ?$ Select all that apply.

A) $12 \vec { i } + 8 \vec { j } - 12 \vec { k }$
B) $- 12 \vec { i } - 8 \vec { j } + 12 \vec { k }$
C) $12 \vec { i } + 2 \vec { j } - 12 \vec { k }$
D) $12 \vec { i } + 8 \vec { j } + 12 \vec { k }$
E) $12 \vec { i } + 2 \vec { j } - 3 \vec { k }$

Question 12

Find the missing numbers:

Question 13

Shortly after takeoff, a plane is climbing with velocity vector Assume the units are kilometer per hour, that the x-axis points east, that the y-axis points north, that the z axis points up.What is the airspeed of the plane?

Question 14

If and , find .

Question 15

A plane wants to go due Northeast, but a wind is blowing toward the North at 78 miles per hour.If the airspeed of the plane is 250 miles per hour, which direction should the plane head? What will be its groundspeed?

Question 16

If , find a unit vector parallel to

Question 17

Let .Considering as a position vector that begins at the origin, at what point of three space does its head lie?

Question 18

If find a value of a such that .

Question 19

If find a unit vector perpendicular to both and

Question 20

If find the value of a making and perpendicular.

Question 21

Consider the plane Find, to three decimal places, the shortest distance from the plane to the point

Question 22

Determine the distance from the plane to the point Give your answer to 4 decimal places.

Question 23

Determine the distance from the plane to the plane .Give your answer to 4 decimal places.
(Hint: Although there are many approaches, projections come in handy.)

Question 24

The angle between $4 \vec { i } + 2 \vec { k }$ and $\vec { i } - 2 \vec { j } + 5 \vec { k }$ is less than $\pi$ /2.

Question 25

The two planes z = -3x - 4y + 5 and -6x - 8y - 2z = 0 are parallel.

Question 26

The plane has a normal Let .Express as the sum of a vector which is parallel to and a vector which is parallel to the plane.

Question 27

Determine the equation of the plane which contains the points (1, 2, -8)and (-2, 1, -8)and is perpendicular to the plane .

Question 28

Two boats leave a harbor at the same time.Boat A cruises northwest at a rate of 17 knots (nautical miles per hour).Boat B cruises north at a rate of 21 knots.
(a)Find the displacement vector from boat A to boat B half an hour later.
(b)For the passengers in boat A, what does the velocity of boat B appear to be?

Question 29

Shortly after takeoff, a plane is climbing with velocity vector Assume the units are kilometer per hour, that the x-axis points east, that the y-axis points north, that the z axis points up.
If there is a wind blowing 20 km/hr to the southeast, what is the speed of the plane relative to the ground?

Question 30

A birdwatcher on the ground notices that a blue bird is heading north with a groundspeed of 28 miles/hr.The wind is blowing 25 miles/hr to the northeast.What is the airspeed of the bird?

Question 31

Find the angle between the planes and .

Question 32

Suppose $\vec { \alpha }$ is a fixed vector of length 5, $\vec { b }$ is a vector which can rotate and has length 2, and $\theta$ is the angle between them.
What is the maximum value of $\vec { a } \cdot \vec { b }$ , and for what value of $\theta$ does it occur?

Question 33

An object, P, is pulled by a force of magnitude 16lb at an angle of 20 degrees north of east.In what direction must a force of magnitude 10 lb pull to ensure that P moves due east?

Question 34

The triangle ABC with vertices $A = \vec { i } + 5 \vec { j }$ , $B = \vec { i } - 3 \vec { j }$ , and $C = - 2 \vec { i } + 2 \vec { j }$ is a right triangle.

Question 35

Suppose and Find values for a and b making perpendicular to both and

Question 36

Consider the plane Find a normal vector to this plane.

Question 37

Find a unit vector in the direction of the vector .

Question 38

Determine a formula for the distance between a point (a, b, c)and the y-axis.

Question 39

Determine two vectors of length 7 in the xy-plane that are normal to the parabola at (2, 4).

Question 40

One of the highlights of the Calaveras Fair, besides the bullfrog jumping contest, is the watermelon seed spitting contest.The seed spitting is being done in the direction of $\vec { s } = 5 \vec { i } + \vec { j }$ .The wind velocity during the contest is $\vec { w } = \vec { i } + 4 \vec { j }$ mph.The rules say that a legal wind in the direction of the seed spit must not exceed 3 mph.Mr.Samuel C.himself just spit for a record of 22 ft 1.4 inches. Will his record stand?

A)Yes
B)No

Question 41

Given the points P = (1, 2, 10), Q = (3, 5, 24), and R = (2, 5, 20), find the equation of the plane containing P, Q, and R.

Question 42

For what value(s)of a is the vector parallel to the plane ?

Question 43

Does the point D(-3, 4, -6)lie in the same plane as the triangle with vertices A(0, 0, 0), B(5, 5, 35), C(5, 4, 33)?

A)No
B)Yes

Question 44

If and , find the value of s so that is perpendicular to .

Question 45

Consider two vectors and .For what values of a are and parallel?

Question 46

Suppose .Find the vector of shortest length with .

Question 47

If $\vec{u}, \vec{v}$ are non-zero vectors with $|\vec{z} \cdot \vec{v}|=\mid \overrightarrow{k z}\|\| \vec{v} \|$ , then $\vec { u }$ is either parallel or anti-parallel to $\vec { v }$ .

Question 48

Compute the area of the triangle with vertices A(0, 0, 0), B(3, 3, 0), and C(3, 6, 3).

Question 49

Given the points P = (1, 2, 9), Q = (3, 5, 26), and R = (2, 5, 22), find the area of the triangle PQR.Give your answer to 4 decimal places.

Question 50

Find the equation of the plane parallel to and and containing the point (1, 1, 2).

Question 51

For what value(s)of a is the vector perpendicular to the plane z = 10x - 2y + 7?

Question 52

Suppose that $\vec { u }$ and $\vec { v }$ are non-zero vectors, and that $\vec { u }$ is perpendicular to $\vec { v }$ .Let $\vec { w } = \vec { u } \times ( \vec { v } \times \vec { u } )$ .Which of the following statements are true? Select all that apply.

A) $\vec { W }$ is perpendicular to $\vec { u }$ ,
B) $\vec { W }$ is perpendicular to $\vec { v }$ ,
C) $\vec { W }$ is parallel to $\vec { u }$ ,
D) $\vec { W }$ is parallel to $\vec { v }$ ,
E) $\vec { W }$ is the zero vector.

Question 53

Suppose and is a vector such that .What is the shortest length can be?

Question 54

Given the points P = (1, 2, 16), Q = (3, 5, 35), and R = (2, 5, 33), find the (perpendicular)distance from the point R to the line through P and Q.Give your answer to 4 decimal places.

Question 55

Let $\vec { u }$ and $\vec { v }$ be two non-zero parallel vectors.Then $\vec { U } \cdot \vec { V } = \pm 1$ .

Question 56

Consider the triangle ABC where A is the point (4, 5, -3), B is the point (0, -3, 2)and C is the point (4, 1, 1).
Find the cosine of angle BAC.Give your answer to 4 decimal places.

Question 57

Vectors $\vec { a } , \vec { b } , \vec { c }$ and $\vec { u }$ are shown in the picture below.

A) $\vec { c } \cdot \vec { u }$ ^< $\vec { b } \cdot \vec { u }$ ^<
$\vec { a } \cdot \vec { u }$
B) $\vec { a } \cdot \vec { u }$ ^< $\vec { c } \cdot \vec { u }$ ^<
$\vec { b } \cdot \vec { u }$
C) $\vec { c } \cdot \vec { u }$ ^<
$\vec { a } \cdot \vec { u }$ ^<
$\vec { b } \cdot \vec { u }$
D) $\vec { b } \cdot \vec { u }$ ^< $\vec { a } \cdot \vec { u }$ ^<
$\vec { c } \cdot \vec { u }$

Question 58

Consider two vectors and .For what value(s)of a are and perpendicular?

Question 59

The vector $\vec { a } \times \vec { b }$ is parallel to the x-axis where $\vec{a}$ and $\vec { b }$ are shown below (both $\vec { \alpha }$ and $\vec { b }$ are in the xy-plane).

Question 60

The vector $\vec { a } \times \vec { b }$ has a positive z component where $\vec{a}$
and $\vec { b }$ are shown below (both $\vec { \alpha }$ and $\vec { b }$ are in the xy-plane).

Question 61

Find the volume of the parallelopiped with edges , , and .

Question 62

The vectors $\vec { v }$ and $\vec { W }$ are shown below.Determine if the following statement is true or false. $( \vec { v } - \vec { w } ) \cdot \vec { j } > 0$

Question 63

Find the vector in 3-space which satisfies both of the following conditions.
(a) (b)

Question 64

Let A(-2, 1, -1), B(-3, 2, 0), C(-2, 1, 1), and D(-1, 0, 0)be the vertices of a parallelogram in that order.
Find the area of the projection of the parallelogram in the xy-plane.

Question 65

Consider the plane .Find the value of a that makes the vector perpendicular to the plane.

Question 66

A surveyor is standing at point A, 403 feet away from a tall building.Let be the direction of east, north and up respectively.She observes that a vector in the direction of is , where C is the highest point of the building.
What is the height of the building?

Question 67

An airplane is descending in preparation to land at an airport to its northwest.Its airspeed is 640 km/hr and its altitude is decreasing a rate of 50 km/hr.Describe its velocity vector.

Question 68

Let ABCDEF be a regular hexagon.Express the vectors and in terms of .

Question 69

Find the missing number.

Question 70

Let ABCDEF be a regular hexagon.Express in terms of .

Question 71

Let and let be a vector of magnitude 4 at an angle of 30° to .Find .

Question 72

Let and .Compute

Question 73

Let ABCDEF be a regular hexagon.Express in terms of .

Question 74

Consider the plane .What is the value of a if the vector is parallel to the plane?

Question 75

Let and let be a vector of magnitude 6 at an angle of 30° to .Find .

Question 76

Consider the plane .Find the coordinates of the points where the plane intersects the three axes.

Question 77

Let A(1, -2, -1), B(-1, -4, 0), C(0, -2, 1), and D(2, 0, 0)be the vertices of a parallelogram in that order.
Find the area of the parallelogram to 4 decimal places.

Question 78

Consider the points P = (1, 2, 6), Q = (3, 4, 4)and R = (-1, 2, 12).
Find a vector perpendicular to and and then use it to find the equation of the plane which contains P, Q, and R.

Question 79

Let and .Compute

Question 80

Find the missing numbers.