C H 6.3 – V ECTORS IN A P LANE

A L ITTLE I NTRO Some quantities are measured as a single real number – area, temperature, time Others involve two quantities – force and velocity These two involve magnitude and direction and are represented as vectors rather than a single quantity.

VECTOR VOCABULARY 1) Directed Line Segment – how to draw a vector, looks like a ray 2) Initial Point – the starting point of the vector 3) Terminal Point – the ending point of the vector 4) Magnitude – the length of a vector Notations P Q

MORE VOCABULARY 5) Standard Position – initial point of a vector at the origin 6) Component Form – when the initial point is at the origin, the vector can be represented by the coordinates of the terminal point, v =, each coordinate is the component of the vector. 7) Zero Vector – both the initial and terminal points of a vector at the origin 8) Unit Vector – a vector with a magnitude of 1

L ET ’ S LOOK AT AN EXAMPLE Let vector w have an initial point at (-7, 4) and a terminal point at (1, -5). Write w in component form then find its length.

E QUALITY For two vectors to be equal, they must have equal magnitudes and equal directions. i.e. u = v, iff (if and only if) u 1 = v 1 and u 2 = v 2 Two vectors can have the same magnitude, but not be equal themselves. Two vectors can have equal directions, but not be equal themselves.

L ET U BE A VECTOR FROM P(-3, -2) TO Q(3, 2). L ET V BE A VECTOR FROM R(-2, 0) TO S(4, 4). S HOW THAT U = V.

AND EVEN MORE VOCABULARY! 9) Scalar Multiplication – multiplying a scalar (a real number) and a vector (or a matrix). The product of a scalar and a vector is a vector. 10) Vector Addition – you can only add vectors to vectors and do this by adding like components. Take 2 vectors, u and v: u v u + v = This is know as the parallelogram law because the resultant is the diagonal of the parallelogram made by u and v.

N EGATIVE V ECTORS Let v be any vector. - v will have the exact same magnitude as v, but will point in the opposite direction. || v || = ||- v || If v =, then –v = v -v

S CALAR M ULTIPLICATION Let k represent a scalar and u represent a vector. What will happen to the resultant of k and u if k is positive? k is negative? |k| is greater than 1? |k| is less than 1?

L ET V = AND W = AND FIND V + 2 W

P ROPERTIES OF VECTOR ADDITION AND SCALAR MULTIPLICATION Let u, v, and w be vectors and c and d be scalars. 1) Vector Addition is associative: 2) Vector Addition is commutative: 3) Zero is the identity of vector addition: 4) The sum of a vector and its opposite is the zero vector: all properties are on p. 448

P ROPERTIES CON ’ T 5) The distributive property is true for distributing a scalar or a vector: 6) You can find the product of all scalars first before multiplying them to a single vector: 7) The identity of scalar multiplication is 1: 8) The resultant of zero and a vector is a scalar (0): 9) The length of c v is the absolute value of c times the length of v :

U NIT V ECTORS Will be useful in future problems They are vectors that point in the same direction as a given vector, but have a magnitude of 1. To find a unit vector: u = You will take each component of v and divide them by the magnitude of v. Vector u is called the unit vector in the direction of v.

F IND THE UNIT VECTOR OF V =

H OMEWORK p. 453 #2, 4, 8, 10, 12, 22, 24, 28, 30

U NIT V ECTORS There are 2 standard unit vectors denoted as i and j. i = The Horizontal Unit Vector j = The Vertical Unit Vector All vectors can be written using these two unit vectors. This is called a linear combination. v = v 1 i + v 2 j v 1 is the horizontal component of the vector v 2 is the vertical component of the vector Example: If v =, then as a linear combination v = 5 i + 8 j

Let u be a vector with an initial point (-4, 3) and a terminal point (2, 9). Write u as a linear combination of i and j.

Let u = -3 i + 6 j and v = 4 i – j. Find 2 u – j.

D IRECTION A NGLES Let u be the unit vector and θ be the angle from the positive x-axis to u (counterclockwise), the terminal point of u is on the unit circle and u = = = (cosθ)i + (sinθ)j If v is any vector that makes and angle θ with the positive x-axis, v = ||v|| = ||v||(cosθ)i + ||v||(sinθ)j = ai + bj

B UT HOW CAN WE FIND Θ ? Recall, tan θ = Therefore, tanθ = = b is the coefficient with j a is the coefficient with i

F IND THE DIRECTION ANGLE OF EACH VECTOR. 1. u = 3 i + 3 j 2. w = 3 i – 4 j REMEMBER TO DOUBLE CHECK THE QUADRANT!

L ET ’ S APPLY THIS TO REAL LIFE ! Velocity A ball is thrown with an initial velocity of 55 feet per second, at an angle of 37 o with the horizontal. Find the vertical and horizontal components of the velocity.

Find the component form of the vector represented by an airplane flying at 110 mph at an angle of 30 o above due west.

W EIGHT – IS IT A SCALAR OR A VECTOR ? A force of 600 pounds is required to pull a boat and a trailer up a ramp inclined at 15 o from the horizontal. Find the combined weight of the boat and trailer.

H OMEWORK p. 454 # 44, 46, 52, 54, 75, 77