Vectors Vector: a quantity that has both magnitude (size) and direction Examples: displacement, velocity, acceleration Scalar: a quantity that has no direction associated with it, only a magnitude Examples: distance, speed, time, mass
Vectors are represented by arrows. The length of the arrow represents the magnitude (size) of the vector. And, the arrow points in the appropriate direction. 50 m/s 20 m/s NW East
Adding vectors graphically • Without changing the length or the direction of any vector, slide the tail of the second vector to the tip of the first vector. 2. Draw another vector, called the RESULTANT, which begins at the tail of the first vector and points to the tip of the last vector.
Adding co-linear vectors(along the same line) B = 4 m A = 8 m A + B = R = 12 m C = 10 m/s D = - 3 m/s C + D = 10 + (-3) = R = 7 m/s
Adding perpendicular vectors How could you find out the length of the RESULTANT? Since the vectors form a right triangle, use the PYTHAGOREAN THEOREM A2 + B2 = C2 11.67 m 6 m 10 m
Vector COMPONENTS Each vector can be described to terms of its x and y components. Y (vertical) component X (horizontal) component If you know the lengths of the x and y components, you can calculate the length of the vector using the Pythagorean.
A boat capable of going 5 m/s in still water is crossing a river with a current of 3 m/s. If the boat points straight across the river, where will it end up- straight across the river or downstream? Downstream, because that is where the current will carry it even as it goes across! What is the resultant velocity?
Sketch the 2 velocity vectors: the boat’s velocity, 5 m/s and the river’s velocity, 3 m/s Resultant velocity, 3 m/s Resultant velocity 5 m/s
Drawing the x and y components of a vector is called “resolving a vector into its components” Make a coordinate system and slide the tail of the vector to the origin. Draw a line from the arrow tip to the x-axis. The components may be negative or positive or zero. X component Y component
Calculating the componentsHow to find the length of the components if you know the magnitude and direction of the vector. Sin q = opp / hyp Cos q = adj / hyp Tan q = opp / adj SOHCAHTOA = 12 m/s A = A sin q Ay = 12 sin 35 = 6.88 m/s q = 35 degrees Ax = A cos q = 12 cos 35 = 9.83 m/s
a x y A B R A = 18, q = 20 degrees B = 15, b = 40 degrees Adding Vectors by components B A b q Slide each vector to the origin. Resolve each vector into its x and y components The sum of all x components is the x component of the RESULTANT. The sum of all y components is the y component of the RESULTANT. Using the components, draw the RESULTANT. Use Pythagorean to find the magnitude of the RESULTANT. Use inverse tan to determine the angle with the x-axis. 18 cos 20 18 sin 20 -15 cos 40 15 sin 40 15.8 5.42 a = tan-1(15.8 / 5.42) = 71.1 degrees above the positive x-axis