Vector
Vectors
Vectors can be graphically represented by directed line segments. The length is selected, according to a few scale, to represent the magnitude of the vector, and the direction of the directed line segment stands for the direction of the vector. For instance, if we allow 1 cm stand for 5 km/h, then a 15-km/h wind from the northwest ought to be represented by a directed line segment 3 cm long, as illustrated in the figure below:
A vector in the plane is a directed line segment. Two vectors are equivalent if they have the same magnitude and direction.
Consider a vector drawn from point A to point B. Point A is known as the initial point of the vector, and point B is known as the terminal point. Symbolic notation for this vector is (read as “vector AB”). Vectors are as well represented by boldface letters such as u, v, and w. The four vectors in the figure at left have the same length and direction. Therefore they stand for equivalent vectors; that is,
In the context of vectors, we make use of = to mean equivalent.
The length, or magnitude, of is denoted as ||. In order to find out whether vectors are equivalent, we obtain their magnitudes and directions.
Example 1
The vectors u, , and w are illustrated in the figure below. Show that u = = w.
Solution
We first calculate the length of each vector with the use of the distance formula:
|u| = √[2 - (-1)]2 + (4 - 3)2 = √9 + 1 = √10,
| | = √[0 - (-3)]2 + [0 - (-1)]2 = √9 + 1 = √10,
|w| = √(4 - 1)2 + [-1 - (-2)]2 = √9 + 1 = √10.
Therefore:
|u| = | = |w|.
The vectors u, , and w seem to go in the same direction. Thus we check their slopes. If the lines that they are on all have the same slope, the vectors have the same direction. We estimate the slopes:
Since u, , and w possess equivalent magnitude and the same direction, u = = w.
Bear in mind that the equivalence of vectors needs just the same magnitude and the same direction and not the same location. In the demonstrations below, each of the first three pairs of vectors are not same. The fourth set of vectors is an example of equivalence.
Suppose that a person takes 4 steps east and then 3 steps north. He or she will then be 5 steps from the starting point in the direction shown. A vector 4 unit long and pointing to the right stands for 4 steps east and a vector 3 unit long and pointing up stands for 3 steps north. The sum of the two vectors is the vector 5 steps in magnitude and in the direction illustrated. The sum is as well known as the resultantof the two vectors.
Generally, two nonzero vectors u and v can be added geometrically by placing the initial point of v at the terminal point of u and then calculating the vector that has the same starting point as u and the same terminal point as v, as illustrated in the following figure.
The sum is the vector demonstrated by the directed line segment from the initial point A of u to the terminal point C of v. That is, if u = and v = , then
u + v = + =
We can as well describe vector addition by putting the initial points of the vectors together, completing a parallelogram, and calculating the diagonal of the parallelogram. This explanation of addition is occasionally known as the parallelogram law of vector addition. Vector addition is commutative. As illustrated in the figure, both u + v and v + u are denoted by the same directed line segment.
If two forces F1 and F2 act on an object, the joined effect is the sum, or resultant, F1 + F2 of the distinct forces.
Example 2
Forces of 15 Newtons and 25 Newtons act on an object at right angles to each other. Calculate their sum, or resultant, giving the magnitude of the resultant and the angle that it makes with the larger force.
Solution
We make a drawing — this time, a rectangle — with the use of v or to represent the resultant. To obtain the magnitude, we make use of the
Pythagoras theorem:
|v|2 = 152 + 252
Here |v| denotes the length, or magnitude, of v.
|v| = √152 + 252
|v| ≈ 29,2.
To obtain the direction, we note that since OAB is a right triangle,
tanθ = 15/25 = 0,6.
With the use of a calculator, we obtain θ, the angle that the resultant makes with the larger force:
θ = tan- 1(0,6) ≈ 31°
The resultant has a magnitude of 29,2 and makes an angle of 31° with the larger force.
Pilots ought to adjust the direction of their flight when there is a crosswind. Both the wind and the aircraft velocities can be explained by vectors.
Example 3
Airplane Speed and Direction
An airplane travels on a bearing of 100° at an airspeed of 190 km/h while a wind is blowing 48 km/h from 220°. Calculate the ground speed of the airplane and the direction of its track, or course, over the ground.
Solution
We at first make a drawing. The wind is represented by and the velocity vector of the airplane by. The resultant velocity vector is v, the sum of the two vectors. The angle θ between v and is known as a drift angle.
Observe that the measure of COA = 100° - 40° = 60°. Therefore, the measure of CBA is as well 60° (opposite angles of a parallelogram are equal). Since the sum of all the angles of the parallelogram is 360° and COB and OAB have the same measure, every one ought to be 120°. By the law of cosines in OAB, we have
|v|2 = 482 + 1902 - 2.48.190.cos120°
|v|2 = 47,524
|v| = 218
Therefore, |v| is 218 km/h. By the law of sines in the same triangle,
48/sinθ = 218/sin120°,
or
sinθ = 48.sin120°/218 ≈ 0,1907
θ ≈ 11°
Therefore, θ = 11°, to the nearest degree. The ground speed of the airplane is 218 km/h, and its track is in the direction of 100° - 11°, or 89°.
Given a vector w, we may want to calculate two other vectors u and v whose sum is w. The vectors u and v are known as of w and the process of calculating them is known as resolving, or representing, a vector into its vector constituents.
When we resolve a vector, we normally look for perpendicular constituents. Most frequently, one constituent will be parallel to the x - axis and the other will be parallel to the y - axis. For this reason, they are frequently known as the horizontal and vertical components of a vector. In the figure below, the vector w = is resolved as the sum of u = and v =.
The horizontal component of w is u and the vertical component is v.
Example 4
A vector w has a magnitude of 130 and is inclined 40° with the horizontal. Resolve the vector into horizontal and vertical components.
Answer
We initially make a drawing illustrating horizontal and vertical vectors u and v whose sum is w.
From ABC, we calculate |u| and |v| with the use of the definitions of the cosine and sine functions:
cos40° = |u|/130, or |u| = 130.cos40° ≈ 100,
sin40° = |v|/130, or |v| = 130.sin40° ≈ 84.
Therefore the horizontal component of w is 100 right, and the vertical component of w is 84 up.
Graphical representation of vectors
Vectors are drawn as arrows. An arrow has both a magnitude (how long it is) and a direction (the direction in which it points). The beginning point of a vector is referred as the tail and the end point is known as the head.
Directions
There are a lot of acceptable methods of writing vectors. In so far as the vector has a magnitude and a direction, it is most probably acceptable. These different methods come from the various methods of representing a direction for a vector.
Relative directions
The easiest way to illustrate direction is with relative directions: to the left, to the right, forward, backward, up and down.
Compass directions
Another common method of showing directions is to make use of the points of a compass: North, South, East, and West. If a vector does not point precisely in one of the compass directions, then we make use of an angle. For instance, we can have a vector pointing 40° North of West. Begin with the vector pointing along the West direction (look at the dashed arrow below), then rotate the vector towards the north until there is a 40° angle between the vector and the West direction (the solid arrow below). The direction of this vector can as well be explained as: W 40° N (West 40° North); or N 50° W (North 50° West).
Bearing
Another method of expressing direction is to make use of a bearing. A bearing is a direction relative to a fixed point. Given just an angle, the rule is to define the angle clockwise with respect to North. Thus, a vector with a direction of 110° has been rotated clockwise 110°> relative to North. A bearing is at all times written as a three digit number, for instance 275° or 080° (for 80°).
Exercise 1:
Scalars and vectors
Problem 1:
Classify the following quantities as scalars or vectors:
1. 12 km
2. 1 m south
3. 2 m•s−1, 45°
4. 075°, 2 cm
5. 100 km•h−1, 0°
Answer 1:
(a) scalar
(b) vector
(c) vector
(d) vector
(e) vector
Question 2:
Make use of two different notations to write down the direction of the vector in each of the following diagrams:
Answer 2:
(a) north; 000º; 360º
(b) E 60º N, N 30º E; 090º
(c) S 40º W, W 50º S; 220º
Drawing vectors
In order to draw a vector correctly we ought represent its magnitude properly and include a reference direction in the diagram. A scale permits us to translate the length of the arrow into the vector's magnitude. For example if one selects a scale of 1 cm = 2 N (1 cm represents 2 N), a force of 20 N towards the East would be denoted as an arrow 10 cm long pointing towards the right. The points of a compass are frequently used to show direction or alternatively an arrow pointing in the reference direction.
Method: Drawing Vectors
1. Choose a scale and write it down.
2. Choose a reference direction
3. Estimate the length of the arrow representing the vector, with the use of the scale.
4. Draw the vector as an arrow. Ensure that you fill in the arrow head.
5. Fill in the magnitude of the vector.
Example 1:
Drawing vectors I
Question
Draw the following vector quantity: = 6 m•s−1 North
Answer
Choose a scale and write it down.
1 cm = 2 m•s−1
Choose a reference direction
Estimate the length of the arrow at the particular scale.
If 1 cm = 2 m•s−1, then 6 m•s−1 = 3 cm
Draw the vector as an arrow.
Scale used: 1 cm = 2 m•s−1
Example 2:
Drawing vectors 2
Question
Draw the following vector quantity: = 16 m east
Answer
Choose a scale and write it down.
1 cm = 4 m
Choose a reference direction
Choose the length of the arrow at the specific scale.
If 1 cm = 4 m, then 16 m = 4 cm
Draw the vector as an arrow
Scale used: 1 cm = 4 m
Direction = East
Vectors
Vectors can be graphically represented by directed line segments. The length is selected, according to a few scale, to represent the magnitude of the vector, and the direction of the directed line segment stands for the direction of the vector. For instance, if we allow 1 cm stand for 5 km/h, then a 15-km/h wind from the northwest ought to be represented by a directed line segment 3 cm long, as illustrated in the figure below:
A vector in the plane is a directed line segment. Two vectors are equivalent if they have the same magnitude and direction.
Consider a vector drawn from point A to point B. Point A is known as the initial point of the vector, and point B is known as the terminal point. Symbolic notation for this vector is (read as “vector AB”). Vectors are as well represented by boldface letters such as u, v, and w. The four vectors in the figure at left have the same length and direction. Therefore they stand for equivalent vectors; that is,
In the context of vectors, we make use of = to mean equivalent.
The length, or magnitude, of is denoted as ||. In order to find out whether vectors are equivalent, we obtain their magnitudes and directions.
Example 1
The vectors u, , and w are illustrated in the figure below. Show that u = = w.
Solution
We first calculate the length of each vector with the use of the distance formula:
|u| = √[2 - (-1)]2 + (4 - 3)2 = √9 + 1 = √10,
| | = √[0 - (-3)]2 + [0 - (-1)]2 = √9 + 1 = √10,
|w| = √(4 - 1)2 + [-1 - (-2)]2 = √9 + 1 = √10.
Therefore:
|u| = | = |w|.
The vectors u, , and w seem to go in the same direction. Thus we check their slopes. If the lines that they are on all have the same slope, the vectors have the same direction. We estimate the slopes:
Since u, , and w possess equivalent magnitude and the same direction, u = = w.
Bear in mind that the equivalence of vectors needs just the same magnitude and the same direction and not the same location. In the demonstrations below, each of the first three pairs of vectors are not same. The fourth set of vectors is an example of equivalence.
Suppose that a person takes 4 steps east and then 3 steps north. He or she will then be 5 steps from the starting point in the direction shown. A vector 4 unit long and pointing to the right stands for 4 steps east and a vector 3 unit long and pointing up stands for 3 steps north. The sum of the two vectors is the vector 5 steps in magnitude and in the direction illustrated. The sum is as well known as the resultantof the two vectors.
Generally, two nonzero vectors u and v can be added geometrically by placing the initial point of v at the terminal point of u and then calculating the vector that has the same starting point as u and the same terminal point as v, as illustrated in the following figure.
The sum is the vector demonstrated by the directed line segment from the initial point A of u to the terminal point C of v. That is, if u = and v = , then
u + v = + =
We can as well describe vector addition by putting the initial points of the vectors together, completing a parallelogram, and calculating the diagonal of the parallelogram. This explanation of addition is occasionally known as the parallelogram law of vector addition. Vector addition is commutative. As illustrated in the figure, both u + v and v + u are denoted by the same directed line segment.
If two forces F1 and F2 act on an object, the joined effect is the sum, or resultant, F1 + F2 of the distinct forces.
Example 2
Forces of 15 Newtons and 25 Newtons act on an object at right angles to each other. Calculate their sum, or resultant, giving the magnitude of the resultant and the angle that it makes with the larger force.
Solution
We make a drawing — this time, a rectangle — with the use of v or to represent the resultant. To obtain the magnitude, we make use of the
Pythagoras theorem:
|v|2 = 152 + 252
Here |v| denotes the length, or magnitude, of v.
|v| = √152 + 252
|v| ≈ 29,2.
To obtain the direction, we note that since OAB is a right triangle,
tanθ = 15/25 = 0,6.
With the use of a calculator, we obtain θ, the angle that the resultant makes with the larger force:
θ = tan- 1(0,6) ≈ 31°
The resultant has a magnitude of 29,2 and makes an angle of 31° with the larger force.
Pilots ought to adjust the direction of their flight when there is a crosswind. Both the wind and the aircraft velocities can be explained by vectors.
Example 3
Airplane Speed and Direction
An airplane travels on a bearing of 100° at an airspeed of 190 km/h while a wind is blowing 48 km/h from 220°. Calculate the ground speed of the airplane and the direction of its track, or course, over the ground.
Solution
We at first make a drawing. The wind is represented by and the velocity vector of the airplane by. The resultant velocity vector is v, the sum of the two vectors. The angle θ between v and is known as a drift angle.
Observe that the measure of COA = 100° - 40° = 60°. Therefore, the measure of CBA is as well 60° (opposite angles of a parallelogram are equal). Since the sum of all the angles of the parallelogram is 360° and COB and OAB have the same measure, every one ought to be 120°. By the law of cosines in OAB, we have
|v|2 = 482 + 1902 - 2.48.190.cos120°
|v|2 = 47,524
|v| = 218
Therefore, |v| is 218 km/h. By the law of sines in the same triangle,
48/sinθ = 218/sin120°,
or
sinθ = 48.sin120°/218 ≈ 0,1907
θ ≈ 11°
Therefore, θ = 11°, to the nearest degree. The ground speed of the airplane is 218 km/h, and its track is in the direction of 100° - 11°, or 89°.
Given a vector w, we may want to calculate two other vectors u and v whose sum is w. The vectors u and v are known as of w and the process of calculating them is known as resolving, or representing, a vector into its vector constituents.
When we resolve a vector, we normally look for perpendicular constituents. Most frequently, one constituent will be parallel to the x - axis and the other will be parallel to the y - axis. For this reason, they are frequently known as the horizontal and vertical components of a vector. In the figure below, the vector w = is resolved as the sum of u = and v =.
The horizontal component of w is u and the vertical component is v.
Example 4
A vector w has a magnitude of 130 and is inclined 40° with the horizontal. Resolve the vector into horizontal and vertical components.
Answer
We initially make a drawing illustrating horizontal and vertical vectors u and v whose sum is w.
From ABC, we calculate |u| and |v| with the use of the definitions of the cosine and sine functions:
cos40° = |u|/130, or |u| = 130.cos40° ≈ 100,
sin40° = |v|/130, or |v| = 130.sin40° ≈ 84.
Therefore the horizontal component of w is 100 right, and the vertical component of w is 84 up.
Graphical representation of vectors
Vectors are drawn as arrows. An arrow has both a magnitude (how long it is) and a direction (the direction in which it points). The beginning point of a vector is referred as the tail and the end point is known as the head.
Directions
There are a lot of acceptable methods of writing vectors. In so far as the vector has a magnitude and a direction, it is most probably acceptable. These different methods come from the various methods of representing a direction for a vector.
Relative directions
The easiest way to illustrate direction is with relative directions: to the left, to the right, forward, backward, up and down.
Compass directions
Another common method of showing directions is to make use of the points of a compass: North, South, East, and West. If a vector does not point precisely in one of the compass directions, then we make use of an angle. For instance, we can have a vector pointing 40° North of West. Begin with the vector pointing along the West direction (look at the dashed arrow below), then rotate the vector towards the north until there is a 40° angle between the vector and the West direction (the solid arrow below). The direction of this vector can as well be explained as: W 40° N (West 40° North); or N 50° W (North 50° West).
Bearing
Another method of expressing direction is to make use of a bearing. A bearing is a direction relative to a fixed point. Given just an angle, the rule is to define the angle clockwise with respect to North. Thus, a vector with a direction of 110° has been rotated clockwise 110°> relative to North. A bearing is at all times written as a three digit number, for instance 275° or 080° (for 80°).
Exercise 1:
Scalars and vectors
Problem 1:
Classify the following quantities as scalars or vectors:
1. 12 km
2. 1 m south
3. 2 m•s−1, 45°
4. 075°, 2 cm
5. 100 km•h−1, 0°
Answer 1:
(a) scalar
(b) vector
(c) vector
(d) vector
(e) vector
Question 2:
Make use of two different notations to write down the direction of the vector in each of the following diagrams:
Answer 2:
(a) north; 000º; 360º
(b) E 60º N, N 30º E; 090º
(c) S 40º W, W 50º S; 220º
Drawing vectors
In order to draw a vector correctly we ought represent its magnitude properly and include a reference direction in the diagram. A scale permits us to translate the length of the arrow into the vector's magnitude. For example if one selects a scale of 1 cm = 2 N (1 cm represents 2 N), a force of 20 N towards the East would be denoted as an arrow 10 cm long pointing towards the right. The points of a compass are frequently used to show direction or alternatively an arrow pointing in the reference direction.
Method: Drawing Vectors
1. Choose a scale and write it down.
2. Choose a reference direction
3. Estimate the length of the arrow representing the vector, with the use of the scale.
4. Draw the vector as an arrow. Ensure that you fill in the arrow head.
5. Fill in the magnitude of the vector.
Example 1:
Drawing vectors I
Question
Draw the following vector quantity: = 6 m•s−1 North
Answer
Choose a scale and write it down.
1 cm = 2 m•s−1
Choose a reference direction
Estimate the length of the arrow at the particular scale.
If 1 cm = 2 m•s−1, then 6 m•s−1 = 3 cm
Draw the vector as an arrow.
Scale used: 1 cm = 2 m•s−1
Example 2:
Drawing vectors 2
Question
Draw the following vector quantity: = 16 m east
Answer
Choose a scale and write it down.
1 cm = 4 m
Choose a reference direction
Choose the length of the arrow at the specific scale.
If 1 cm = 4 m, then 16 m = 4 cm
Draw the vector as an arrow
Scale used: 1 cm = 4 m
Direction = East
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