In this section you will: Evaluate the sum of two or more vectors in two dimensions graphically. Solve for the sum of two or more vectors mathematically. Section 5.1-1

Vectors in Multiple Dimensions When vectors do not point along the same straight line we must add them mathematically or graphically. You need a protractor, to draw the vectors at the correct angle and to measure the direction and magnitude of the resultant vector. Vectors can be added by placing them tip-to-tail and then drawing the resultant of the vector by connecting the tail of the first vector to the tip of the second vector. Section 5.1-2

Vectors in Multiple Dimensions The figure below shows the two forces in the free-body diagram. Section 5.1-3

Vectors in Multiple Dimensions Move one of the vectors so that its tail is at the same place as the tip of the other vector. Section 5.1-4

Vectors in Multiple Dimensions The resultant vector is drawn by pointing from the tail of the first vector to the tip of the last vector and measure it to find its magnitude. Use a protractor to measure the direction of the resultant vector. Section 5.1-5

Vectors in Multiple Dimensions Two or more vectors (A, B, C, etc.) may be added by first resolving each vector into its x- and y-components. The x-components are added to form the x-component of the resultant: Rx = Ax + Bx + Cx. Section 5.1-26

Vectors in Multiple Dimensions Similarly, the y-components are added to form the y-component of the resultant: Ry = Ay + By + Cy. Section 5.1-27

Vectors in Multiple Dimensions The Pythagorean theorem can be used to solve for the length or a resultant vector produced from a right triangle. R = the magnitude of the resultant vector R B A Section 5.1-6

Vectors in Multiple Dimensions The angle can be found with trigonometry. tan-1(Opposite/Adjacent) = ϴ In this example tan-1(B/A) = ϴ R X ϴ A Section 5.1-6

Finding the Magnitude of the Sum of Two Vectors Mathematically When the angle is 90°, use the Pythagorean theorem to find the magnitude of the resultant vector. Section 5.1-14

Find total displacement for a motion that go 15 km in one direction then turns and goes 25 km at a right angle to the initial direction. Because displacement is a vector, if you solve for the magnitude and angle of the resultant you will get the displacement. R 25 ϴ 15

Finding the Magnitude of the Sum of Two Vectors Mathematically Substitute A = 25 km, B = 15 km Section 5.1-14

Finding the Magnitude of the Sum of Two Vectors Mathematically Tan-1(Opp/Adj) = ϴ Tan-1(25/15) = 59o Make sure you set your calculator into the degrees setting when doing trigonometry in physics!!! Section 5.1-14

Question 1 Jeff moved 3 m due north, and then 4 m due west to his friend’s house. What is the displacement of Jeff? A. 3 + 4 m B. 4 – 3 m C. 32 + 42 m D. 5 m Section 5.1-31

Answer 1 Reason: When two vectors are at right angles to each other as in this case, we can use the Pythagorean theorem of vector addition to find the magnitude of the resultant, R. Section 5.1-32

Answer 1 Reason: The Pythagorean theorem of vector addition states If vector A is at a right angle to vector B, then the sum of squares of magnitudes is equal to the square of the magnitude of the resultant vector. That is, R2 = A2 + B2  R2 = (3 m)2 + (4 m)2 = 5 m Section 5.1-33