Adding vectors. Let s consider some vectors to be added.

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1 Vectors Some physical quantities have both size and direction. These physical quantities are represented with vectors. A common example of a physical quantity that is represented with a vector is a force.

2 Vectors Some physical quantities have both size and direction. These physical quantities are represented with vectors. A common example of a physical quantity that is represented with a vector is a force. For example, consider the gravitational force that acts on an object. This gravitational force, commonly known as the weight of the object, has size and direction. If an object weighs 100 N, the size of the gravitational force acting on that object is 100 N, and the direction is towards the center of the earth.

3 Vectors Some physical quantities have both size and direction. These physical quantities are represented with vectors. A common example of a physical quantity that is represented with a vector is a force. For example, consider the gravitational force that acts on an object. This gravitational force, commonly known as the weight of the object, has size and direction. If an object weighs 100 N, the size of the gravitational force acting on that object is 100 N, and the direction is towards the center of the earth. Because vectors have both size and direction, we cannot just add vectors like we add numbers. We need special rules for adding vectors.

4 Adding vectors Let s consider some vectors to be added.

5 Adding vectors Let s consider some vectors to be added. Consider a vector A that is 50 N that is in a direction of 20 above the +x direction. 20 A = 50N

6 Adding vectors Let s consider some vectors to be added. Consider a vector A that is 50 N that is in a direction of 20 above the +x direction. Consider another vector B that is 30 N in a direction of 30 to the left of the +y direction. B = 30N 30 A = 50N 20

7 Adding vectors using the tip-to-tail convention Start by drawing one of the vectors

8 Adding vectors using the tip-to-tail convention Start by drawing one of the vectors 20 A

9 Adding vectors using the tip-to-tail convention Start by drawing one of the vectors Then, add the other vector by placing its tail at the tip of the first vector. B A

10 Adding vectors using the tip-to-tail convention Start by drawing one of the vectors Then, add the other vector by placing its tail at the tip of the first vector. Finally, draw the resultant vector R from the tail of the first vector to the tip of the second vector. R B A

11 Adding vectors using the tip-to-tail convention Note that you get the same resultant vector if you start with the other vector. Here we start with vector B.

12 Adding vectors using the tip-to-tail convention Note that you get the same resultant vector if you start with the other vector. Here we start with vector B. B 30

13 Adding vectors using the tip-to-tail convention Note that you get the same resultant vector if you start with the other vector. Here we start with vector B. Then, add in A by placing its tail at the tip of the B. A 20 B 30

14 Adding vectors using the tip-to-tail convention Note that you get the same resultant vector if you start with the other vector. Here we start with vector B. Then, add in A by placing its tail at the tip of the B. Finally, draw the resultant vector R from the tail of the B to the tip of A. A 20 R B 30

15 Breaking vectors into components Although it is possible to find the resultant of adding two vectors using triangles and the Law of Cosines and/or the Law of Sines, we will use the rectangular component method.

16 Breaking vectors into components Although it is possible to find the resultant of adding two vectors using triangles and the Law of Cosines and/or the Law of Sines, we will use the rectangular component method. Breaking vectors into rectangular components is very useful for several types of engineering analysis, including Truss analysis.

17 Breaking vectors into components Although it is possible to find the resultant of adding two vectors using triangles and the Law of Cosines and/or the Law of Sines, we will use the rectangular component method. Breaking vectors into rectangular components is very useful for several types of engineering analysis, including Truss analysis. In addition, using the rectangular components method is faster than using triangles and the Law of Cosines and Law of Sines, if you are adding more than two vectors together to form the resultant vector.

18 Getting the rectangular components of a vector Coming up with the rectangular components of a vector involves drawing a line from the tip of the vector. This line is drawn parallel to either the x-axis or the y-axis, whichever is more convenient. A 20

19 Getting the rectangular components of a vector Coming up with the rectangular components of a vector involves drawing a line from the tip of the vector. This line is drawn parallel to either the x-axis or the y-axis, whichever is more convenient. Here we draw a line parallel to the y-axis A 20

20 Getting the rectangular components of a vector Coming up with the rectangular components of a vector involves drawing a line from the tip of the vector. This line is drawn parallel to either the x-axis or the y-axis, whichever is more convenient. Here we draw a line parallel to the y-axis Then, we draw a line from the tail parallel to the other axis. In this case, we draw a line from the tail parallel to the x-axis A 20

21 Getting the rectangular components of a vector Coming up with the rectangular components of a vector involves drawing a line from the tip of the vector. This line is drawn parallel to either the x-axis or the y-axis, whichever is more convenient. Here we draw a line parallel to the y-axis Then, we draw a line from the tail parallel to the other axis. In this case, we draw a line from the tail parallel to the x-axis A 20 A y A x The red line is parallel to the y-axis and is considered the y-component of the vector. The blue line is parallel to the x-axis and is considered the x-component of the vector.

22 Getting the rectangular components of a vector In the last example, the triangle chosen to obtain the rectangular components made use of the fact that the angle was known with respect to the x-axis. Consider the following vector B B 30

23 Getting the rectangular components of a vector In the last example, the triangle chosen to obtain the rectangular components made use of the fact that the angle was known with respect to the x-axis. Consider the following vector B Here we draw a line parallel to the x-axis B 30

24 Getting the rectangular components of a vector In the last example, the triangle chosen to obtain the rectangular components made use of the fact that the angle was known with respect to the x-axis. Consider the following vector B Here we draw a line parallel to the x-axis Then, we draw a line from the tail parallel to the other axis. In this case, we draw a line from the tail parallel to the y-axis B 30

25 Getting the rectangular components of a vector In the last example, the triangle chosen to obtain the rectangular components made use of the fact that the angle was known with respect to the x-axis. Consider the following vector B Here we draw a line parallel to the x-axis Then, we draw a line from the tail parallel to the other axis. In this case, we draw a line from the tail parallel to the y-axis B x B 30 B y The red line is the y-component, and the blue-line is the x-component.

26 Using trigonometry to calculate rectangular components Now we can use trigonometry to solve for the size of those rectangular components.

27 Using trigonometry to calculate rectangular components Now we can use trigonometry to solve for the size of those rectangular components. Here is a right triangle with sine and cosine defined. θ a c sin(θ) = opposite hypotenuse = b c cos(θ) = adjacent hypotenuse = a c b

28 Breaking A into rectangular components Here is vector A. A = 50N 20

29 Breaking A into rectangular components Here is vector A. This is how to calculate the components. A = 50N 20 A x A y sin(20 ) = Ay A

30 Breaking A into rectangular components Here is vector A. This is how to calculate the components. A = 50N 20 A x A y sin(20 ) = Ay A Ay = +Asin(20 ) = +50 sin(20 ) = N.

31 Breaking A into rectangular components Here is vector A. This is how to calculate the components. A = 50N 20 A x A y sin(20 ) = Ay A Ay = +Asin(20 ) = +50 sin(20 ) = N. cos(20 ) = Ax A

32 Breaking A into rectangular components Here is vector A. This is how to calculate the components. A = 50N 20 A x A y sin(20 ) = Ay A Ay = +Asin(20 ) = +50 sin(20 ) = N. cos(20 ) = Ax A Ax = +Acos(20 ) = +50 cos(20 ) = N.

33 Breaking B into rectangular components Here is vector B. B = 30N 30

34 Breaking B into rectangular components Here is vector B. Note that the sine function is associated with the component on the opposite side of the known angle. B x B = 30N 30 Bx = Bsin(30 ) = 30 sin(30 ) = 15.0N.

35 Breaking B into rectangular components Here is vector B. Note that the sine function is associated with the component on the opposite side of the known angle. Similarly the cosine function is associated with the component on the adjacent side to the known angle. B x B = 30N 30 B y Bx = Bsin(30 ) = 30 sin(30 ) = 15.0N. By = +Bcos(30 ) = +30 cos(30 ) = N.

36 Adding A to B Now we can add A to B by combining the components in the x-direction, and then the components in the y-direction.

37 Adding A to B Now we can add A to B by combining the components in the x-direction, and then the components in the y-direction. R x = A x +B x = = 31.99N

38 Adding A to B Now we can add A to B by combining the components in the x-direction, and then the components in the y-direction. R x = A x +B x = = 31.99N R y = A y +B y = = 43.08N

39 Adding A to B Now we can add A to B by combining the components in the x-direction, and then the components in the y-direction. R x = A x +B x = = 31.99N R y = A y +B y = = 43.08N We can draw these components to obtain the resultant vector

40 Adding A to B Now we can add A to B by combining the components in the x-direction, and then the components in the y-direction. R x = A x +B x = = 31.99N R y = A y +B y = = 43.08N We can draw these components to obtain the resultant vector R x = 31.99N

41 Adding A to B Now we can add A to B by combining the components in the x-direction, and then the components in the y-direction. R x = A x +B x = = 31.99N R y = A y +B y = = 43.08N We can draw these components to obtain the resultant vector R y = 43.08N R x = 31.99N

42 Adding A to B Now we can add A to B by combining the components in the x-direction, and then the components in the y-direction. R x = A x +B x = = 31.99N R y = A y +B y = = 43.08N We can draw these components to obtain the resultant vector R R y = 43.08N R x = 31.99N

43 Obtaining the size (magnitude) of the resultant vector To get the size (magnitude) of the vector, we use the Pythagorean theorem. R = Rx 2 +Ry 2 R = = 53.66N

44 Obtaining the direction of the resultant vector To get the direction of the resultant vector, we can make use of the trigonometric function, tangent.

45 Obtaining the direction of the resultant vector To get the direction of the resultant vector, we can make use of the trigonometric function, tangent. For a right triangle, tangent is defined in this way: θ a tan(θ) = opposite adjacent = b a c b

46 Obtaining the direction of the resultant vector To get the direction of the resultant vector, we can make use of the trigonometric function, tangent. For a right triangle, tangent is defined in this way: θ a tan(θ) = opposite adjacent = b a To find the angle, you just take the inverse function ( ) b θ = tan 1 a c b

47 Getting the direction for R R R y = 43.08N θ R x = 31.99N tan(θ) = R y = R x ( ) θ = tan 1 = R = 53.66N, above the +x-axis

Unit 2: Trigonometry. This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses.

Unit 2: Trigonometry. This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses. Unit 2: Trigonometry This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses. Pythagorean Theorem Recall that, for any right angled triangle, the square