Vectors and Projectiles - Detailed Help

Assignment VP4: Adding Right Angle Vectors

Objectives:
  • The student should be able to use the Pythagorean theorem to add two or more vectors in order to determine the magnitude of the resultant.
  • The student should be able to trigonometric principles to add two or more vectors in order to determine the direction of the resultant.

 

Reading:

The Physics Classroom, Vectors and Motion in Two Dimensions Unit, Lesson 1, Part a

The Physics Classroom, Vectors and Motion in Two Dimensions Unit, Lesson 1, Part b

The Physics Classroom, Vectors and Motion in Two Dimensions Unit, Lesson 1, Part c

 

 

A city jogger runs 13 blocks due east and then 21 blocks due north. The magnitude of the jogger's displacement is ____ blocks. Enter a numerical answer accurate to the second decimal place. (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


The head-to-tail method of vector addition can be used to create a rough sketch of this physical situation. The first vector (13 blocks, east) is sketched (not to scale) in its indicated direction. The second vector (21 blocks, north) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. The Pythagorean Theorem can then be used to calculate the magnitude of the resultant (see Math Magic section).


Pythagorean Theorem

When two vectors which make a right angle to each other are added together, the resultant vector is the hypotenuse of a right triangle. The Pythagorean theorem can be used to calculate the magnitude of the resultant. If the right triangle has sides with lengths of x and y, then the length of the hypotenuse is the square root of the sum of the squares of the sides. That is,
hypotenuse = SQRT (x2 + y2)

An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section above). Consider the following steps:

  • Sketch the first vector in the appropriate direction. Place an arrowhead at the end of the vector.
  • Starting at the arrowhead of the first vector, draw the second vector in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Since the resultant is a hypotenuse of a right triangle, the Pythagorean theorem can be used to calculate its magnitude (see Math Magic section above).

How can the Pythagorean theorem be used to determine the magnitude of the resultant of two right angle vectors?

 

 

An ant walks 3.3 meters due west and then 1.7 meters due north. The magnitude of the ant's displacement is ___ meters. Enter a numerical answer accurate to the second decimal place. (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


The head-to-tail method of vector addition can be used to create a rough sketch of this physical situation. The first vector (3.3 meters, west) is sketched (not to scale) in its indicated direction. The second vector (1.7 meters, north) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. The Pythagorean Theorem can then be used to calculate the magnitude of the resultant (see Math Magic section).


Pythagorean Theorem

When two vectors which make a right angle to each other are added together, the resultant vector is the hypotenuse of a right triangle. The Pythagorean theorem can be used to calculate the magnitude of the resultant. If the right triangle has sides with lengths of x and y, then the length of the hypotenuse is the square root of the sum of the squares of the sides. That is,
hypotenuse = SQRT (x2 + y2)

An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section above). Consider the following steps:

  • Sketch the first vector in the appropriate direction. Place an arrowhead at the end of the vector.
  • Starting at the arrowhead of the first vector, draw the second vector in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Since the resultant is a hypotenuse of a right triangle, the Pythagorean theorem can be used to calculate its magnitude (see Math Magic section above).

How can the Pythagorean theorem be used to determine the magnitude of the resultant of two right angle vectors?

 

 

A student in 1982 was playing the interactive video game known as Pac-man. In this game he moved a yellow Pac-man 17 cm up and 12 cm left. The Pac-man's net displacement was closest to ___ cm. Enter a numerical answer accurate to the second decimal place. (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


The head-to-tail method of vector addition can be used to create a rough sketch of this physical situation. The first vector (17 cm, up) is sketched (not to scale) in its indicated direction. The second vector (12 cm, left) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. The Pythagorean Theorem can then be used to calculate the magnitude of the resultant (see Math Magic section).


Pythagorean Theorem

When two vectors which make a right angle to each other are added together, the resultant vector is the hypotenuse of a right triangle. The Pythagorean theorem can be used to calculate the magnitude of the resultant. If the right triangle has sides with lengths of x and y, then the length of the hypotenuse is the square root of the sum of the squares of the sides. That is,
hypotenuse = SQRT (x2 + y2)

An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section above). Consider the following steps:

  • Sketch the first vector in the appropriate direction. Place an arrowhead at the end of the vector.
  • Starting at the arrowhead of the first vector, draw the second vector in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Since the resultant is a hypotenuse of a right triangle, the Pythagorean theorem can be used to calculate its magnitude (see Math Magic section above).

How can the Pythagorean theorem be used to determine the magnitude of the resultant of two right angle vectors?

 

 

In the Vector Addition Lab, the following data was collected for determining the displacement from the door of the Physics classroom to another location in the building: 2 m, West; 14 m, South; 22 m, East; 19 m, North; and 2 m, West. The magnitude of the resultant displacement from the physics classroom to the assigned location is closest to ... . (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section below). Consider the following steps:

  • Begin by simplifying the collection of small displacements by adding all the east-west vectors together. Consider west to be the negative direction and add the negative westward displacements to the positive eastward displacements. Record the result and label as E-W (for sum of the east and west vectors).
  • Repeat the process of adding the north-south vectors together, considering south to be the negative direction. Record the result and label as N-S (for sum of the north and south vectors). The two vector sums will now be added together.
  • Consider the sum of the east-west vectors ( E-W) as a single vector and sketch it in the appropriate direction. Place an arrowhead at the end of the vector.
  • Consider the sum of the north-south vectors ( N-S) as a single vector, Starting at the arrowhead of the first vector, draw this second vector in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Since the resultant is a hypotenuse of a right triangle, the Pythagorean theorem can be used to calculate its magnitude (see Math Magic section below).

The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. In this method, the first vector (the E-W vectors) is sketched (not to scale) in its indicated direction. The second vector (the N-S vectors) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. The Pythagorean Theorem can then be used to calculate the magnitude of the resultant (see Math Magic section below).


Pythagorean Theorem

When two vectors which make a right angle to each other are added together, the resultant vector is the hypotenuse of a right triangle. The Pythagorean theorem can be used to calculate the magnitude of the resultant. If the right triangle has sides with lengths of x and y, then the length of the hypotenuse is the square root of the sum of the squares of the sides. That is,
hypotenuse = SQRT (x2 + y2)

How can the Pythagorean theorem be used to determine the magnitude of the resultant of two right angle vectors?

 

 

An ant on a picnic table travels 18 cm eastward, then 20 cm northward and finally 12 cm westward. What is the magnitude of the ant's displacement (in cm)? (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section below). Consider the following steps:

  • Begin by simplifying the east and the west displacements by adding these two vectors together. Consider west to be the negative direction and add the negative westward displacement to the positive eastward displacement. Record the result and label as E-W (for sum of the east and west vectors).
  • Consider the sum of the east-west vectors ( E-W) as a single vector and sketch it in the appropriate direction. Place an arrowhead at the end of the vector.
  • Starting at the arrowhead of the first vector, draw this second vector (the northward vector) in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Since the resultant is a hypotenuse of a right triangle, the Pythagorean theorem can be used to calculate its magnitude (see Math Magic section below).

The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. In this method, the first vector (the E-W vectors) is sketched (not to scale) in its indicated direction. The second vector (20 cm, north) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. The Pythagorean Theorem can then be used to calculate the magnitude of the resultant (see Math Magic section below).


Pythagorean Theorem

When two vectors which make a right angle to each other are added together, the resultant vector is the hypotenuse of a right triangle. The Pythagorean theorem can be used to calculate the magnitude of the resultant. If the right triangle has sides with lengths of x and y, then the length of the hypotenuse is the square root of the sum of the squares of the sides. That is,
hypotenuse = SQRT (x2 + y2)

How can the Pythagorean theorem be used to determine the magnitude of the resultant of two right angle vectors?

 

 

While on a boy scout hike, Dexter walked 6 km north, 3 km east, 5 km north, and 11 km west. The resultant displacement for Dexter's hike is ... . (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section below). Consider the following steps:

  • Begin by simplifying the collection of small displacements by adding all the east-west vectors together. Consider west to be the negative direction and add the negative westward displacements to the positive eastward displacements. Record the result and label as E-W (for sum of the east and west vectors).
  • Repeat the process of adding the north-south vectors together, considering south to be the negative direction. Record the result and label as N-S (for sum of the north and south vectors). The two vector sums will now be added together.
  • Consider the sum of the east-west vectors ( E-W) as a single vector and sketch it in the appropriate direction. Place an arrowhead at the end of the vector.
  • Consider the sum of the north-south vectors ( N-S) as a single vector, Starting at the arrowhead of the first vector, draw this second vector in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Since the resultant is a hypotenuse of a right triangle, the Pythagorean theorem can be used to calculate its magnitude (see Math Magic section below).

The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. In this method, the first vector (the E-W vectors) is sketched (not to scale) in its indicated direction. The second vector (the N-S vectors) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. The Pythagorean Theorem can then be used to calculate the magnitude of the resultant (see Math Magic section below).


Pythagorean Theorem

When two vectors which make a right angle to each other are added together, the resultant vector is the hypotenuse of a right triangle. The Pythagorean theorem can be used to calculate the magnitude of the resultant. If the right triangle has sides with lengths of x and y, then the length of the hypotenuse is the square root of the sum of the squares of the sides. That is,
hypotenuse = SQRT (x2 + y2)

How can the Pythagorean theorem be used to determine the magnitude of the resultant of two right angle vectors?

 

 

A city jogger runs 6 blocks due east and then 11 blocks due north. The direction of the jogger's displacement is ____ degrees. (Use the counterclockwise from East convention.) (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. In this method, the first vector (6 blocks, east) is sketched (not to scale) in its indicated direction. The second vector (11 blocks, north) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. A trigonometric function can then be used to calculate the angle which the axis makes with one of the the nearest axes (see Math Magic section below).


SOH CAH TOA

The trigonometric functions sine, cosine and tangent can be used to express the relationship between the angle of a right triangle and the lengths of the adjacent side, opposite side and hypotenuse. The meaning of the three functions are:
sine = (length of opposite side / length of hypotenuse)

cosine = (length of adjacent side / length of hypotenuse)

tangent = (length of opposite side / length of adjacent side)


An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section above). Consider the following steps:

  • Sketch the first vector in the appropriate direction. Place an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Starting at the arrowhead of the first vector, draw the second vector in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Label the angle between the resultant vector and the horizontal leg of the triangle as angle theta ().
  • Since the two vectors being added and the resultant form a right triangle, SOH CAH TOA can be used to calculate the angle (see Math Magic section above). The tangent function can be used to relate the angle to the length of the horizontal and vertical legs of this right triangle.
  • If the angle is the angle between the resultant R and the east direction, then is the answer to the question. If it is not the angle between east and the resultant, then you will have to use and your sketch to determine the direction of the vector (see the Define Help section below).

Counterclockwise from East Convention for Vector Direction

The direction of a vector is often expressed using the counterclockwise (CCW) convention. According to this convention, the direction of a vector is the number of degrees of rotation which the vector makes counterclockwise from East.

How can the trigonometric functions be used to determine the direction of the resultant of two right angle vectors?

How can the counterclockwise from East convention be used to determine the direction of a vector?

 

 


An ant walks 4.8 meters due west and then 2.9 meters due north. The direction of the ant's displacement is ____ degrees. (Use the counterclockwise from East convention.) (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


There are a several conventions for expressing the direction of a vector. The convention used here is the counterclockwise (CCW) from East convention (see Define Help section below). It is possible that a student could do all his/her math correctly but miss the question because he/she failed to use the CCW convention.


The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. In this method, the first vector (4.8 meters, west) is sketched (not to scale) in its indicated direction. The second vector (2.9 meters, north) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. A trigonometric function can then be used to calculate the angle which the axis makes with one of the the nearest axes (see Math Magic section below).


SOH CAH TOA

The trigonometric functions sine, cosine and tangent can be used to express the relationship between the angle of a right triangle and the lengths of the adjacent side, opposite side and hypotenuse. The meaning of the three functions are:
sine = (length of opposite side / length of hypotenuse)

cosine = (length of adjacent side / length of hypotenuse)

tangent = (length of opposite side / length of adjacent side)

An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section above). Consider the following steps:

  • Sketch the first vector (4.8 meters, west) in the appropriate direction. Place an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Starting at the arrowhead of the first vector, draw the second vector (2.9 meters, north) in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Label the angle between the resultant vector and the horizontal leg of the triangle as angle theta ().
  • Since the two vectors being added and the resultant form a right triangle, SOH CAH TOA can be used to calculate the angle (see Math Magic section above). The tangent function can be used to relate the angle to the length of the horizontal and vertical legs of this right triangle.
  • The angle is the angle between the resultant R and the west direction (if you have followed the strategy described above). Thus, is NOT the answer to the question since you are to state the direction using the counterclockwise from East convention (see the Define Help section below). Since is the angle between west and the resultant, the angle between the resultant and east as measured counterclockwise is 180 degrees - .

Counterclockwise from East Convention for Vector Direction

The direction of a vector is often expressed using the counterclockwise (CCW) convention. According to this convention, the direction of a vector is the number of degrees of rotation which the vector makes counterclockwise from East.

How can the trigonometric functions be used to determine the direction of the resultant of two right angle vectors?

How can the counterclockwise from East convention be used to determine the direction of a vector?

 

 

In the Vector Addition Lab, the following data was collected for determining the displacement from the door of the Physics classroom to another location in the building: 2 m, West; 14 m, South; 22 m, East; 19 m, North; and 2 m, West. The direction of the resultant displacement is closest to ___ degrees. (Note: Numbers are randomized numbers and likely different from the numbers listed here.)


There are a several conventions for expressing the direction of a vector. The convention used here is the counterclockwise (CCW) from East convention (see Define Help section below). It is possible that a student could do all his/her math correctly but miss the question because he/she failed to use the CCW convention.


An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It section below). Consider the following steps:

  • Begin by simplifying the collection of small displacements by adding all the east-west vectors together. Consider west to be the negative direction and add the negative westward displacements to the positive eastward displacements. Record the result and label as E-W (for sum of the east and west vectors).
  • Repeat the process of adding the north-south vectors together, considering south to be the negative direction. Record the result and label as N-S (for sum of the north and south vectors). The two vector sums will now be added together.
  • Sketch the first vector (the E-W vector) in the appropriate direction. Place an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Starting at the arrowhead of the first vector, draw the second vector (the N-S vector) in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • At the tail of the resultant, label the angle between the resultant vector and the adjacent leg as the angle theta ().
  • Since the two vectors being added and the resultant form a right triangle, SOH CAH TOA can be used to calculate the angle (see Math Magic section below). The tangent function can be used to relate the angle to the lengths of the horizontal and vertical legs of this right triangle.
  • The angle is the angle between the resultant R and the adjacent leg. Thus, is NOT necessarily the answer to the question since you are to state the direction using the counterclockwise from East convention (see the Define Help section below). Since is the angle between the resultant and one of the axes, the angle between the resultant and east as measured counterclockwise can be determined using the value of .

The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. The sum of the east-west vectors and the north-south vectors can be added together. In this method, the first vector (the E-W vector) is sketched (not to scale) in its indicated direction. The second vector (the N-S vector) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. A trigonometric function can then be used to calculate the magnitude of the resultant (see Math Magic section below).


SOH CAH TOA

The trigonometric functions sine, cosine and tangent can be used to express the relationship between the angle of a right triangle and the lengths of the adjacent side, opposite side and hypotenuse. The meaning of the three functions are:
sine = (length of opposite side / length of hypotenuse)

cosine = (length of adjacent side / length of hypotenuse)

tangent = (length of opposite side / length of adjacent side)

Counterclockwise from East Convention for Vector Direction

The direction of a vector is often expressed using the counterclockwise (CCW) convention. According to this convention, the direction of a vector is the number of degrees of rotation which the vector makes counterclockwise from East.

How can the trigonometric functions be used to determine the direction of the resultant of two right angle vectors?

How can the counterclockwise from East convention be used to determine the direction of a vector?