Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print, an italic letter will be used: A

Chapter 4 Vector Addition Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected

Chapter 4 Vector Addition Negative Vectors Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions) A = -B Resultant Vector The resultant vector is the sum of a given set of vectors

Chapter 4 Vector Addition When adding vectors, their directions must be taken into account Units must be the same Graphical Methods Use scale drawings Algebraic Methods More convenient

Chapter 4 Vector Addition The resultant is the sum of two or more vectors. Vectors can be added by moving the tail of one vector to the head of another vector without changing the magnitude or direction of the vector. Vector Addition Note: The red vector R has the same magnitude and direction.

Chapter 4 Vector Addition Multiplying a vector by a scalar number changes its length but not its direction unless the scalar is negative. V 2V -V

Chapter 4 Vector Addition If two vectors are added at right angles, the magnitude can be found by using the Pythagorean Theorem R2 = A2 + B 2 and the angle by If two vectors are added at any other angle, the magnitude can be found by the Law of Cosines and the angle by the Law of Sines

Chapter 4 Vector Addition 8 meters 62+82=102 36° 6 meters 10 meters The distance traveled is 14 meters and the displacement is 10 meters at 36º south of east.

Chapter 4 Vector Addition A hiker walks 3 km due east, then makes a 30° turn north of east walks another 5 km. What is the distance and displacement of the hiker? The distance traveled is 3 km + 5 km = 8 km R2 = 32+52- 2*3*5*Cos 150° R2 = 9+25+26=60 R = 7.7 km R 5 km 3 km The displacement is 7.7 km @ 19° north of east

Chapter 4 Vector Addition Add the following vectors and determine the resultant. 3.0 m/s, 45 and 5.0 m/s, 135 5.83 m/s, 104

Chapter 4 Vector Addition

Chapter 4 Vector Addition A boat travels at 30 m/s due east across a river that is 120 m wide and the current is 12 m/s south. What is the velocity of the boat relative to shore? How long does it take the boat to cross the river? How far downstream will the boat land? 30 m/s 30 m/s 12 m/s 12 m/s = 32. 3 m/s @ 21° downstream. The speed will be The time to cross the river will be t = d/v = 120 m / 30 m/s = 4 s The boat will be d = vt = 12 m/s * 4 s = 48 m downstream.

Chapter 4 Vector Addition Examples

Chapter 4 Vector Addition

Chapter 4 Vector Addition

Chapter 4 Vector Addition Add the following vectors and determine the resultant. 6.0 m/s, 225 + 2.0 m/s, 90 4.80 m/s, 207.9 

Chapter 4 Vector Addition Add the following vectors and determine the resultant. 6.0 m/s, 225 + 2.0 m/s, 90 R2 = 22 + 62 – 2*2*6*cos 45 R2 = 4 + 36 –24 cos 45 R2 = 40 – 16.96 = 23 R = 4.8 m R 6 m 2 m 45° R = 4.8 m @ 208 

Chapter 4 Vector Addition A component is a part It is useful to use rectangular components These are the projections of the vector along the x- and y-axes

Chapter 4 Vector Addition The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis Then,

Chapter 4 Vector Addition The previous equations are valid only if θ is measured with respect to the x-axis The components can be positive or negative and will have the same units as the original vector The components are the legs of the right triangle whose hypotenuse is A May still have to find θ with respect to the positive x-axis

Chapter 4 Vector Addition Choose a coordinate system and sketch the vectors Find the x- and y-components of all the vectors Add all the x-components This gives Rx:

Chapter 4 Vector Addition Add all the y-components This gives Ry: Use the Pythagorean Theorem to find the magnitude of the Resultant: Use the inverse tangent function to find the direction of R:

Chapter 4 Vector Addition Vector components is taking a vector and finding the corresponding horizontal and vertical components. Vector resolution A Ay Ax

Chapter 4 Vector Addition A plane travels 500 km at 60°south of east. Find the east and south components of its displacement. de de= 500 km *cos 60°= 250 km 60° ds ds= 500 km *sin 60°= 433 km 500 km