9-6 Vectors

A vector is a quantity that has both magnitude, or length, and direction, and is represented by a directed segment. Symbol: Sketch:

Component Form *A vector is in standard position if its initial point is at the origin. We can represent vectors by an ordered pair with brackets. Called the Component Form. <change in x, change in y>, from the tip to the tail of the directed segment.

EX) Write the component form of the vector with initial point A(-1, -4) and end point B(-5, -1)

vector with initial point A(-1, -4) and end point B(-5, -1) To find magnitude, we use the distance formula to calculate the length of the vector.  What’s the magnitude of the above vector?

vector with initial point A(-1, -4) and end point B(-5, -1) Direction of a vector is found by creating a right triangle and using some trig!

Vectors are equal if they are equal in magnitude (length) and direction. Vectors are parallel iff they have the same or opposite direction

Vectors can be used to move points ** Vectors can be used to move points. Just add each component (x and y) to the chosen point. EX) Give a mapping of the image of quadrilateral HJLK with vertices H(-4, 4), J(-2, 4), L(-1, 2), and K(-3, 1) under the translation v = <5, -5>

** Vectors can be combined to perform a composition of translations by adding the vectors’ corresponding components. If

A vector can also be multiplied by a scalar (a number), that will change the magnitude, but not the direction.

Solve Problems Using Vectors Ex) Suppose a person is canoeing due east across a river at 4 miles per hour.  a) If the river is flowing south at 3 miles per hour, what are the resultant direction and speed of the canoe?

b) If the current is reduced by half of its original speed, what are the resultant direction and speed of the canoe?