Trigonometry and Vectors Motion and Forces in Two Dimensions SP1b. Compare and constract scalar and vector quantities.

Standard SP1. Students will analyze the relationships between force, mass, gravity, and the motion of objects. b. Compare and contrast scalar and vector quantities. SP1b. Compare and constract scalar and vector quantities.

Right Triangles The longest side is the hypotenuse. It is opposite the 90 º angle. The other two sides are named depending on where they are in relation to the angle you are looking at. SP1b. Compare and constract scalar and vector quantities.

The Pythagorean Theorem For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger. SP1b. Compare and constract scalar and vector quantities.

Animation SP1b. Compare and constract scalar and vector quantities.

Formula The formula we use is: a 2 + b 2 = c 2 C = the length of the hypotenuse The other two sides are a and b. SP1b. Compare and constract scalar and vector quantities.

We Also Need to Know the Angles SP1b. Compare and constract scalar and vector quantities. Acute angle A is drawn in standard position as shown. Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position,

Easy Way To Remember SP1b. Compare and constract scalar and vector quantities.

Example Find the values of sin A, cos A, and tan A in the right triangle. Solution – length of side opposite angle A is 7 – length of side adjacent angle A is 24 – length of hypotenuse is 25

VECTORS AND VECTOR RESOLUTION SP1b. Compare and constract scalar and vector quantities.

Scalar SP1b. Compare and constract scalar and vector quantities.

Vector SP1b. Compare and constract scalar and vector quantities.

Vectors SP1b. Compare and constract scalar and vector quantities.

Vector Addition VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started. SP1b. Compare and constract scalar and vector quantities.

Vector Subtraction VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started. SP1b. Compare and constract scalar and vector quantities.

More Examples SP1b. Compare and constract scalar and vector quantities.

Angle Direction SP1b. Compare and constract scalar and vector quantities.

Vectors Are Typically Drawn to Scale SP1b. Compare and constract scalar and vector quantities.

So How Do We Add These? SP1b. Compare and constract scalar and vector quantities.

PYTHAGOREAN THEOREM SP1b. Compare and constract scalar and vector quantities.

Example SP1b. Compare and constract scalar and vector quantities.

Example Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement. SP1b. Compare and constract scalar and vector quantities.

PARALLELOGRAM METHOD SP1b. Compare and constract scalar and vector quantities.

You Might Find this Useful SP1b. Compare and constract scalar and vector quantities.

USING TRIG FUNCTIONS Determining Direction SP1b. Compare and constract scalar and vector quantities.

Vectors can be broken down into components A fancy way of saying how far it goes on the x and on the y. We calculate the sides using trig.

"A" is used to represent the vector. SP1b. Compare and constract scalar and vector quantities.

Now the Other Side SP1b. Compare and constract scalar and vector quantities.

Example What if you have a vector that is o ? SP1b. Compare and constract scalar and vector quantities.

So How Do We Find Value of Direction? The direction is given as an angle from the +x axis Positive is counterclockwise SP1b. Compare and constract scalar and vector quantities.

GRAPHICAL METHOD (TIP TO TAIL) SP1b. Compare and constract scalar and vector quantities.

Graphical Method SP1b. Compare and constract scalar and vector quantities.

The Order Doesn’t Matter Same three vectors, different order SP1b. Compare and constract scalar and vector quantities.

Animation SP1b. Compare and constract scalar and vector quantities.

Resultants SP1b. Compare and constract scalar and vector quantities.

(100 km/hr) 2 + (25 km/hr) 2 = R km 2 /hr km 2 /hr 2 = R km 2 /hr 2 = R 2 SQRT( km 2 /hr 2 ) = R km/hr = R SP1b. Compare and constract scalar and vector quantities.

tan (theta) = (opposite/adjacent) tan (theta) = (25/100) theta = invtan (25/100) theta = 14.0 degrees SP1b. Compare and constract scalar and vector quantities.

Animation SP1b. Compare and constract scalar and vector quantities.

Steps in vector addition 1.Sketch the vector – Head to tail method – Parallelogram method 2.Break vectors into their components 3. Add the components to calculate the components of the resultant vector 4.Calculate the magnitude of R 5.Calculate the direction of R Add 180 o to  if R x is negative SP1b. Compare and constract scalar and vector quantities.

Example A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. SP1b. Compare and constract scalar and vector quantities.

Example A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. SP1b. Compare and constract scalar and vector quantities.