6.3 Vectors in the Plane

I. Vectors.

A) Vector = a segment that has both length and direction.

1) Symbol = AB or a bold lower case letter v.

2) Draw a ray on the coordinate plane for a vector.

B) In vector AB, A is the initial pt. and B is the terminal pt.

1) It starts at point A and stops at point B.

C) Magnitude = the length of the vector. (Symbol = )

1) It’s the distance formula. Use the initial & terminal pts.

2) Symbol =

2122

12 yyxxd

AB

v

6.3 Vectors in the Plane

II. Vectors can be …

A) Same direction vectors (or parallel vectors).

1) Vectors with the same direction (think parallel lines).

B) Equal vectors.

1) Vectors that have the same slope and magnitude.

C) Opposite vectors.

1) Vectors that have the same magnitude but go in

opposite directions.

D) Vectors can represent distances traveled (2 miles North) or forces acting upon an object (canoe paddled at 5 mph across a river with a current of 7 mph).

6.3 Vectors in the Plane

III. Component Form of a Vector.A) Standard position = a vector whose initial point is at (0 , 0).

1) It is represented by its terminal point (x , y) using the

symbol v = or

B) Magnitude = or simply

C) Any vector can be put into standard position by …

1) Shifting it: Move the initial point to the origin (0 , 0) but

keep its same magnitude & direction & you will get its

terminal point .

2) Or by doing math: AB = = v

a) Subtraction order is important (terminal – initial)

yx, 21,vv

v 22

21 vv 22 yx

21,vv

ABAB yyxx ,

6.3 Vectors in the Plane

IV. Resultants.A) Resultant = the sum of two or more vectors.

1) Symbol = r

B) Finding resultants.

1) Tip to tail method: put the starting point of the 2nd

vector at the terminal point of the 1st vector.

a) Also known as the parallelogram method because

the resultant is the diagonal of the parallelogram

formed from the 2 vectors. (only good for 2 vectors)

2) Or shift the vectors to the origin (component form).

a) v + u = (xv + xu , yv + yu) =

b) Also called Vector Addition (or sum of vectors).

Homework page 456 # 1 – 14 all

21,vv