CHAPTER 3 Section 2 Vector Operations

Objectives◦ Identify appropriate coordinate systems for solving problems with vectors

◦ Apply the Pythagorean theorem and tangent function to calculate the magnitude and direction of the resultant vector

◦ Resolve vectors into components using the sine and cosine functions

◦ Add vectors that are not perpendicular

Coordinate systems in two dimensions ◦ In you remember in chapter 2 we look at the motion of a gecko and how the gecko

movement was describe a as motion along the x-axis. The movement was just one dimension

◦Now lets see at a plane’s movement

Coordinate Systems in two dimensions◦Now what happens with the plane . Its that we are adding another axis, the y-axis

as well as the x-axis.to helps us describe the motion in two dimension and simplifies analysis of motion in one dimension.

◦One problem is that if we use the first method the axis must be turn again if the direction of the plane changes.

◦ Another problem is that first method provides no way of to deal with a second airplane that is not traveling on the same direction as the first plane.

Determining Resultant Magnitude and direction ◦ In the previous section we found out the magnitude and direction of the resultant

graphically.

◦However , this approach is time consuming, and the accuracy of the answer depends on how carefully the diagram is drawn and measured.

◦Now a simpler method is to use the Pythagorean theorem and the tangent function to find the magnitude and direction of the resultant vector.

Pythagorean Theorem

The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle. It is use to find the magnitude of the resultant vector

(hypotenuse)2 (leg 1)2 (leg 2)2

Tangent Function

We use the tangent function to find the direction of the resultant vector.

tangent of angle = opposite leg

adjacent leg

Example #1◦ An archaeologist climbs the great pyramid in Giza, Egypt. The pyramid’s height is

136 m and its width 2.30 x10^2 m. What is the magnitude and the direction of the displacement of the archeologist after she has climbed from the bottom of the pyramid to the top?

Problems from book◦ Let’s work problems 1-4

Example #2◦ Convert the vector(5,7)into magnitude/angle format

Example#3◦ Convert the vector(13,13)into magnitude/angle form.

Example#4◦ Convert the vector(-1.0,1.0) into magnitude/angle form

Resolving vectors into components◦What are the components of a vector?

◦ The projections of a vector along the axes of a coordinate system

◦ The y component is parallel to the y –axis and the x component is parallel to the x-axis.

◦ Resolving a vector allows you to analyze the motion in each direction.

◦ The sine and cosine can be used to find the components of a vector.

sine of angle = opposite leg

hypotenuse

cosine of angle = adjacent leg

hypotenuse

Example #5◦ Find the components of the velocity of a helicopter traveling 95 km/h at a angle of

35 degrees to the ground?

Example #6◦ Add the two vectors. One has a magnitude 5 and angle 45, and the other has a

magnitude 7 and angle 35.

Example #7◦ Add a vector whose magnitude is 13 and angle is 27b to one whose magnitude is

11 and angle is 45.

Example #8◦ Add two vectors: Vector one has a magnitude 22 and angle of 19, and vector two

has a magnitude 19 and angle of 48.

Closure◦ Today we continue and look more deeply into vectors. Next class we are going to

look at projectile motion.