Competency Goal 4: The learner will develop an understanding of forces and Newton's Laws of Motion Assess the independence of the vector components of forces.

The Components of a Vector Suppose a car moves along a straight line from start to finish. The displacement vector is shown by r. However, the car could also arrive at the finish by first moving due east, then turning 90 o, and then moving due north.

The Components of a Vector The vectors x and y are called the x and y vector components of r. Components are the horizontal and vertical parts of a vector.

The Components of a Vector The components x and y, when added vectorally, convey exactly the same meaning as does the original vector r. They indicate how the finish point is displaced relative to the starting point.

The Components of a Vector In general, the components of any vector can be used in place of the vector itself in any calculation where it is convenient to do so.

The Components of a Vector The other feature of vector components is that x and y are not just any two vectors added together to give the original vector r … … they are perpendicular vectors. This perpendicular characteristic is a valuable asset in problem solving, as we will soon see.

The Components of a Vector Any type of vector may be expressed in terms of its components, in a way similar to that illustrated.

The Components of a Vector There are times when this drawing is not the most convenient way to represent vector components. An alternate method: The disadvantage of this method is that the head-to-tail arrangement is missing.

DEFINITION OF VECTOR COMPONENTS In two dimensions, the vector components of vector A are two perpendicular vectors A x and A y that are parallel to the x and y axes, respectively, and add together vectorally so that A = A x + A y

SCALAR COMPONENTS It is often easier to work with scalar components, A x and A y (note that the italics symbols), rather than the vector components A x and A y. Scalar components are positive or negative numbers (with units).

RESOLVING A VECTOR INTO ITS COMPONENTS If the magnitude and direction of a vector are known, it is possible to find the components of the vector. The process of finding the components of a vector is called “resolution.” As the next example illustrates, the process can be carried out with the aid of trigonometry because the two perpendicular components always form a right triangle.

Example – Finding the Components of a Vector A displacement vector r has a magnitude of r = 175 m and points at an angle of 50.0 o relative to the x axis. Find the x and y components. Reasoning We will base our solution on the fact that the triangle formed by vector r and its components x and y is a right triangle. Therefore, we can use the sine and cosine functions from trig.

Example – Finding the Components of a Vector A displacement vector r has a magnitude of r = 175 m and points at an angle of 50.0 o relative to the x axis. Find the x and y components. Solution 1 The y component can be obtained using the 50.0 o angle and sin θ = y/r: y = r sin θ = (175 m)(sin 50.0 o ) = 134 m In a similar way, the x component can be found using cos θ = x/r: x = r cos θ = (175 m )(sin 50.0 o ) = 112 m

Example – Finding the Components of a Vector A displacement vector r has a magnitude of r = 175 m and points at an angle of 50.0 o relative to the x axis. Find the x and y components. Solution 2 The angle α can also be used to find the components. Since α o = 90.0 o, it follows that α = 40.0 o. The solution using α yields the same answers as in Solution 1: cos α = y/r y = r cos α = (175 m)(cos 40.0 o ) = 134 m sin α = x/r: x = r sin α = (175 m )(sin 40.0 o ) = 112 m

RESOLVING A VECTOR INTO ITS COMPONENTS Problem solving insight: It is possible for one of the components of a vector to be zero. This does not mean that the vector itself is zero, however. For a vector to be zero, every vector component must be individually be zero.

RESOLVING A VECTOR INTO ITS COMPONENTS Problem solving insight: Two vectors are equal if, and only if, they have the same magnitude and direction. Thus, if one displacement vector points east and another points north, they are not equal, even if they have the same magnitude of 480 m.

Vector Addition by Means of Components The components of a vector provide the most convenient and accurate way of adding (or subtracting) any number of vectors.

Vector Addition by Means of Components For example, suppose that vector A is added to vector B. The resultant is C, where C = B + A.

Vector Addition by Means of Components In part (b), the vector B x has been shifted down and arranged head-to-tail with A x. Similarly, B y and A y have been added head-to- tail.

Vector Addition by Means of Components The x components are colinear … and add together to give the x component of resultant C.

Vector Addition by Means of Components Similarly, the y components are colinear … and add together to give the y component of resultant C.

Vector Addition by Means of Components In terms of scalar components, we can write C 2 = C x 2 + C y 2 The angle θ that C makes with the x axis is given by θ = tan -1 (C y /C x ).

Example Vector Addition by Components A jogger runs 145 m in a direction 20.0 o east of north (displacement vector A) and then 105 m in a direction 35.0 o south of east (displacement vector B). Determine the magnitude and direction of the resultant vector C for these two displacements.

Example Vector Addition by Components Reasoning First, we need to find the components of A and B. We will add all of the components of A and B to find the components of C. Finally, we will use the Pythagorean theorem and trig to find C.

Example Vector Addition by Components Solution The first two rows of the table below give the x and y components of the vectors A and B. Note that the component of B will be negative, since B y points south. Vector x Component y Component A A x = (145 m) sin 20.0 o = 49.6 m A y = (145 m) cos 20.0 o = 136 m B B x = (105 m) cos 35.0 o = 86.0 m B y = (105 m) sin 35.0 o = m C C x = Ax + B x = m C y = A y +B y = 76 m

Example Vector Addition by Components The third row of the table gives the x and y components of the resultant vector C. Part (b) of the drawing shows C and its vector components.

Example Vector Addition by Components The magnitude of C is found using the Pythagorean theorem as C = √C x 2 + C y 2 = √ (135.6 m) 2 + (76 m) 2 = 155 m C makes an angle with the x axis by θ = tan -1 (76 m / m) = 29 o

You have finished the vector components powerpoint. So finally, the goal & objective again:

Competency Goal 4: The learner will develop an understanding of forces and Newton's Laws of Motion Assess the independence of the vector components of forces.