Section 6.3 Vectors 1. The student will represent vectors as directed line segments and write them in component form 2. The student will perform basic operations on vectors. 3. The student will write vectors as linear combinations of unit vectors. 4. The student will find direction angles of vectors.

Vector – a quantity that has both magnitude (length) and direction Vectors can be denoted by boldface, lowercase letters or arrow notation with the arrow above the terminal point. Q (terminal point) P (initial point)

The component form of the vector with initial point P = and the terminal point Q = is The magnitude is denoted by and can be found by using the distance formula

Two vectors are equal if they have the same magnitude and the same direction(slope) Show that u=v v=(0,0) (4,1) u=(2,4) (6,5)

Ex.1 Find component form and magnitude of vector v that has initial point (3, - 6) and terminal point ( - 2, 8). Sketch the vector. Ex. 2 Find component form and magnitude of vector v that has initial point (- 3, - 5) and terminal point ( 5, 1). Sketch the vector.

Vector Addition If u = and v = then u + v = To add vectors geometrically position them (without changing their lengths or directions) so that the initial point of one coincides with the terminal point of the other. The sum u + v (often called the resultant) is formed by joining the initial point of the first vector with the terminal point of the second vector.

the negative of is (i.e. turn in opposite direction) the difference of u and v is u - v = u + -v (i.e. flip v and add to u) scalar multiple of k times u is the vector ku =

ex. Given, find the following algebraically & geometrically a. v + w b. – 2v c. 3w – v

The vector u is called a unit vector in the direction of v if ex. find the unit vector in the direction of and verify that the result has a magnitude of 1.

The unit vectors are called the standard unit vectors and are denoted by linear combination of a vector is denoted by. The scalars are called the horizontal and vertical components of v, respectively.

ex. Let u be the vector with initial point (3, -5) and terminal point (-2, 4). Write u as a linear combination of the standard unit vectors i and j. ex. Let u be the vector with initial point (-1, -5) and terminal point (2, 3). Write u as a linear combination of the standard unit vectors i and j.

Find a unit vector in the direction of Find the vector v with the given magnitude and the same direction as u. (Hint: = magnitude (unit vector))

ex. Let u = -2i + 8j and v = 2i –j. Find a) u + v b) 2u – v c) 3v –2u

If u is a unit vector such that is the angle (measured counterclockwise) from the positive x-axis to u, the terminal point of u lies on the unit circle and you have The angle is the direction angle of the vector u.

ex. Find the direction angle of each vector. a. 5i + 5jb. 4i – 3j c. –2i + 3j

Find the magnitude and direction for each of the following: A) B)

(PreAP only) Find the component form of the sum of u and v with direction angles

ex. Find the component form of the vector that represents the velocity of an airplane descending at a speed of 100 miles per hour at an angle of 40 degrees below the negative horizontal axis.