What you should learn: Represent vectors as directed line segments. Write the component forms of vectors. Perform basic vector operations and represent them graphically. Write vectors as linear combinations of unit vectors. Find the direction angles of vectors. Use vectors to model and solve real-life problems.

Quantities that involve both magnitude and direction can’t be expressed by a single real number. Therefore, you need the concept of vectors. Force and velocity are examples of vectors.

Vector – A line segment with both direction and magnitude. Magnitude – The length of a vector.

  Initial point Terminal point

Two directed line segments that have the same magnitude and direction are equivalent.

Vectors are represented by lower case, boldface letters. Usually use the letters u, v and w

Let u be the vector from P(0,0) to Q(3,2). Let v be the vector from R(1,2) to S(4,4). Show that u and v are equivalent.

Standard Position of a Vector- A vector whose initial point is the origin.

Component Form of a Vector- Given initial point P(p 1,p 2 ) and terminal point Q(q 1,q 2 ), the component form would be… v = PQ =  q 1 -p 1,q 2 -p 2 

Magintude of a Vector- The length of the vector can be found by using the Pythagorean Theorem/Distance Formula. v =  (q 1 -p 1 ) 2 + (q 2 -p 2 ) 2

Component Form of a Vector Vector Animation

x-component of a vector - drop a line from the tip of the original vector straight down to the x-axis and draw a vector along the x- axis from the origin to where this line hits the x- axis, then this newly drawn vector is the x-component of the original vector Component Form of a Vector

y-component of a vector - drop a line from the tip of the original vector straight across to the y-axis and draw a vector along the y- axis from the origin to where this line hits the y- axis, then this newly drawn vector is the y-component of the original vector Component Form of a Vector

If v = 1, then v is called a unit vector. If v = 0, then v is called the zero vector.

Find the component form and the magnitude of the vector, v, with initial point (-3,2) and terminal point (1,4). Interpret what your solution means.

What if I know the magnitude and the direction the vector is going. How can I find its component form? Component Form of a Vector  M x y sin  = yMyM cos  = xMxM So x = Mcos  and y = Msin  Component form =  Mcos , Msin 

Find the component form of the vector that has a magnitude of 30 at an angle of 120 .

 Scalar Multiplication  Vector Addition

Looks like… 2 v or -3 v or 3 / 4 v What does it do? It changes the length and often times the direction of the vector.

v 2v2v Changed the length, but not the direction… the length was doubled.

v 1/2v1/2v Changed the length, but not the direction… the length was cut in half.

v Changed the length, and the direction… the length multiplied by 2 and the direction reversed. -2v

How does it affect the component form? If v =  u 1,u 2  then k v = k  u 1,u 2  =  ku 1,ku 2 

Looks like… u + v What does it do? It creates a resultant vector.

Position the two vectors you are trying to add without changing their lengths or directions. u + v u v Rearrange the vectors so that the initial point of the second vector touches the terminal point of the first vector. Then create a vector by connecting the initial point of u to the terminal point of v. This is called the RESULTANT vector. u + v

u v w u + v + w

How does it affect the component form? If u =  x 1,y 1  and v =  x 2,y 2  then u + v =  x 1 +x 2,y 1 +y 2 

Addition of Vectors and Component Form

Looks like… u - v What does it do? It creates a resultant vector. You need to think of it as… u + (- v)

Position the two vectors you are trying to subtract without changing their lengths or directions. u – v or u + (-v) u v Think what negative v would look like. -v Rearrange the vectors so that the initial point of the second vector touches the terminal point of the first vector. Then create a vector by connecting the initial point of u to the terminal point of -v. This is called the RESULTANT vector.

How does it affect the component form? If u =  x 1,y 1  and v =  x 2,y 2  then u - v =  x 1 -x 2,y 1 -y 2 

 There exists a zero vector.  A vector of magnitude one is called a unit vector.  A vector A multiplied by a scalar m is a vector, unchanged in direction, but modified in length by the factor m.  The negative of a vector is the original vector flipped 180 degrees.  Two vectors, A and B, are added by placing the tail of one on the tip of the other (in either order) and defining the sum to be the vector drawn. from the tail of the first to the tip of the second.  A vector B can be subtracted from a vector A by adding -B to A. Summary of Vectors

Given: u =  3,4  ; v =  6,-2  ; w =  -2,1  Find: u + v 3 v – 2 w w – u + 4 v

The Dot Product of Vectors does not result in another vector. Instead, it results in a scalar value (a single number). If u =  x 1,y 1  and v =  x 2,y 2  then u  v = x 1 x 2 +y 1 y 2

Given: u =  3,4  ; v =  6,-2  ; w =  -2,1  Find: w v  w u  v + u

Where is this useful? It is needed to find the angle between two vectors.

cos  = v  u v u

If v = 1, then v is called a unit vector. i =  1,0  which is 1 unit long and rests along the x axis. j =  0,1  which is 1 unit long and rests along the y axis.

To find a unit vector with the same direction as a given vector, we divide by the magnitude of the vector. For example, consider the vector v = 1, 3 which has a magnitude of. If we divide each component of v by we will get the unit vector u v which is in the same direction as v. Unit Vectors

Given: u =  3,4  ; v =  6,-2  ; w =  -2,1  Find: A unit vector that is in the same direction as w. ( u w ) Describe v using the basic unit vectors, i & j.