12.2 Vectors in the Plane Component form Vector Operations Linear combination of standard unit vectors Force and Velocity

Scalar vs. Vector Temperature in this room at 7:40AM February 3, 2015 Total mass of the observable universe Force pressing on a table exerted by a person leaning on it with her elbow (points in direciton in which the force acts and its length is a measure of the force’s strength) Velocity of an asteroid heading toward Earth (need to know where its headed and how fast it’s going Tension Displacement (need to know direction an object has moved and how far)

Vector w is in “standard form” since its initial point is at the origin. Two vectors are equal iff their standard position is identical.

More about vectors Two nonzero vectors u and v are parallel if there is some scalar c such that u=cv. Parallel vectors can be used to show collinearity. Vectors that go in opposite directions can also be considered parallel!!!

Ex: Find the component form and magnitude of the vector v that has (1, 7) as its initial point and (4, 3) as its terminal point. Component form:

Scalar Multiples of Vectors = Parallel Vectors

Associative Law Geometric proof of (u+v)+w=u+(v+w)

Let u= and v=. Sketch the operations geometrically. Then find: (a) 3v (b) u+v

Unit Vector: must have same direction but a magnitude of 1. Find a unit vector in the direction of v =. Then, check to see if your unit vector really has a magnitude of one. Note: Speed if the magnitude of velocity.

Find the vector v with the given magnitude and same direction as u. ||v|| = 3, u =

Basis Vectors

Linear combination

Let u be the vector with initial point (2, -5) and terminal points (-1,3), and let v=2i-j. Write each of the vectors as a linear combination of i and j. a.u b.w=2u-3v

Find the component form of v and sketch the specified vector operations geometrically u= 2i-j w=i+2j v=-u+2w

How it relates to Geometry Show that the triangle with vertices given by P = (2, 1, 6), Q = (4, 7, 9), and R = (8, 5, -6) is a right triangle. (Use the fact that a right triangle must satisfy the Pythagorean Theorem

Can extend the distance formula to determine if a triangle is isosceles or right triangles! Consider the triangle whose corners are the points P = (1, 1,-1), Q = (2, -1, 1), R = (4, 1, 2) Is the triangle isosceles or right?

Force Problem A small cart is being pulled along a smooth horizontal floor with a 20-lb force F making a 45 degree angle to the floor. What is the “effective” force moving the cart forward? Note: effective force is the horizontal component of F.

The airplane heads due east at 200 mph through a crosswind blowing due north at 30 mph, and the superposition (i.e. sum) of these two velocities is the airplane’s actual velocity vector. What is the airplane’s actual speed and direction? Velocity/Navigation Problem

Answer:

Navigation #2 (Bearing Question) A plane is traveling due east at 200 km/hr. A tailwind at a pi/4 radians north of east is going 40 mph. What is the resultant speed of the plane?

A 50 kg block is suspended by 2 identical cables of the same length forming 45 degree angles with the horizontal. If the block remains motionless over time, then what is the magnitude of the tensile force exerted by each of the cables on the block? Forces acting on an object Note: Since given 50 kg, you must calculate the Force due to gravity!

Free body diagram Answer:

May the force be with you! Calculate the magnitude of the force on cables 1 and 2.

HOMEWORK: SECTION 12.2 Classwork/Homework