Vectors Vector: a quantity that has both magnitude (size) and direction Examples: displacement, velocity, acceleration Scalar: a quantity that has no.

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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

Draw a line from the arrow tip to the x-axis. 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 components How 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 Ay = A sin q = 12 sin 35 = 6.88 m/s q = 35 degrees Ax = A cos q = 12 cos 35 = 9.83 m/s

Adding Vectors by components A = 18, q = 20 degrees B = 15, b = 40 degrees Adding Vectors by components B A a b 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. q A B R x y 18 cos 20 18 sin 20 -15 cos 40 15 sin 40 5.42 15.8 a = tan-1(15.8 / 5.42) = 71.1 degrees above the positive x-axis

Unit Vectors A unit vector is a vector that has a magnitude of exactly 1 unit. Depending on the application, the unit might be meters, or meters per second or Newtons or… The unit vectors are in the positive x, y, and z axes and are labeled

Examples of Unit Vectors A position vector (or r = 3i + 2j ) is one whose x-component is 3 units and y-component is 2 units (usually meters). A velocity vector The velocity has an x-component of 3t units (it varies with time) and a y-component of -4 units (it is constant). Usually the units are meters per second.

Working with unit vectors Suppose the position, in meters, of an object was given by r = 3t3i + (-2t2 - 4t)j What is v? Take the derivative of r! What is a? Take the derivative of v! What is the magnitude and direction of v at t = 2 seconds? Plug in t = 2, pythagorize i and j, then use arc tan to find the angle!

Vector Multiplication Scalar product (Vector)(scalar) = vector Example: Force (a vector): F = ma “dot” product vector • vector = scalar Example: Work (a scalar): W = F • d “cross” product vector x vector = vector Example: Torque (a vector): t = r x F

Magnitude of Cross products: A x B = ABsinq Dot products: A • B = ABcosq Magnitude of Cross products: A x B = ABsinq Use the “right-hand rule” to determine the direction of the resultant vector. Multiplication using unit vector notation….

Direction of cross products i x j = k j x k = i k x i = j j x i = -k k x j = -i i x k = -j + ijkijk -

You can only DOT vectors that have colinear components! 3i • 4i = 12 (NOT i - dot product yield scalars) 3i x 4i = 0 (3)(4)cos 0 = 12 (3)(4) sin 0 = 0 You can only CROSS vectors that have perpendicular components. 3i • 4j = 0 3i x 4j = 12k (k because cross products yield vectors) (3)(4)cos 90 = 0 (3)(4)sin 90 = 12