Vectors and Scalars Physics Scalar A SCALAR is

Vectors and Scalars Physics

Scalar A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it. Magnitude – A numerical value with units. Scalar Example Magnitude Speed 20 m/s Distance 10 m Age 15 years Heat 1000 calories

Vector A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION. Vector Velocity Magnitude & Direction 20 m/s, N Acceleration 10 m/s/s, E Force 5 N, West Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude.

Applications of Vectors VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. n Example: A man walks 54. 5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? 54. 5 m, E + 84. 5 m, E 30 m, E Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION.

Applications of Vectors VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. n Example: A man walks 54. 5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54. 5 m, E 30 m, W 24. 5 m, E -

Non-Collinear Vectors When 2 vectors are perpendicular, you must use the Pythagorean theorem. A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT. The hypotenuse in Physics Finish is called the RESULTANT. 55 km, N Horizontal Component Vertical Component 95 km, E Start The LEGS of the triangle are called the COMPONENTS

BUT……what about the direction? In the previous example, DISPLACEMENT was asked for and since it is a VECTOR we should include a DIRECTION on our final answer. N W of N E of N N of E N of W N of E E W NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. S of W S of E W of S E of S S

BUT…. . what about the VALUE of the angle? ? ? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. 109. 8 km To find the value of the angle we use a Trig function called TANGENT. 55 km, N q N of E 95 km, E So the COMPLETE final answer is : 109. 8 km, 30 degrees North of East

What if you are missing a Suppose a person walked 65 m, 25 degrees East of North. What component? were his horizontal and vertical components? H. C. = ? V. C = ? 25 65 m The goal: ALWAYS MAKE A RIGHT TRIANGLE! To solve for components, we often use the trig functions since and cosine.

Example A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. - 12 m, W - = 6 m, S = 23 m, E 14 m, N 20 m, N 35 m, E 14 m, N R q 23 m, E The Final Answer: 26. 93 m, 31. 3 degrees NORTH or EAST

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. 8. 0 m/s, W 15 m/s, N Rv q The Final Answer : 17 m/s, @ 28. 1 degrees West of North

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. H. C. =? 32 63. 5 m/s V. C. = ?

Identifying Direction Write the angles shown below by using references to east, south, west, north. N N W 50 o S 500 S of E 45 o E W E S 450 W of N

Vectors and Polar Coordinates Polar coordinates (R, q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example. 90 o 40 m 180 o R 180 o 50 o 0 o 270 o R is the magnitude and q is the direction. q 0 o 270 o

2 D Coordinate Spaces n n All that really matters are the numbers. The abstract version of this is called a 2 D Cartesian coordinate space. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 15

From the right triangle in Figure 3. 2 b, we find that Sin θ = y/r and that cos θ = x/r. Therefore, starting with the plane polar coordinates of any point, we can obtain the Cartesian coordinates by using the equations 2/25/2021 17

Example 2 (Cont. ): A 400 -m walk in a direction of 30 o N of E. How far is the displacement east and how far north? N Note: x is the side adjacent to angle 300 400 m y=? 30 o E ADJ = HYP x Cos 300 x=? x = R cos q x = (400 m) cos 30 o = +346 m, E The x-component is: Rx = +346 m

Example 2 (Cont. ): A 400 -m walk in a direction of 30 o N of E. How far is the displacement east and how far north? N Note: y is the side opposite to angle 300 400 m y=? 30 o E OPP = HYP x Sin 300 x=? y = R sin q y = (400 m) sin 30 o = + 200 m, N The y-component is: Ry = +200 m

Finding Resultant: (Cont. ) Finding (R, q) from given (x, y) = (+40, -30) 40 lb Rx q f R= tan f = Ry R 30 lb x 2 + y 2 -30 40 R= 40 lb 30 lb (40)2 + (30)2 = 50 lb f = -36. 9 o q = 323. 1 o

R q Ry f 40 lb R = 50 lb Rx 40 lb Rx q q 30 lb R Ry Rx 40 lb Rx f f Ry 30 lb R q 30 lb Four Quadrants: (Cont. ) R = 50 lb 40 lb Ry R 30 lb f = 36. 9 o; q = 36. 9 o; 143. 1 o; 216. 9 o; 323. 1 o

Unit vector notation (i, j, k) y j k z Consider 3 D axes (x, y, z) i x Define unit vectors, i, j, k Examples of Use: 40 m, E = 40 i 40 m, W = -40 i 30 m, N = 30 j 30 m, S = -30 j 20 m, out = 20 k 20 m, in = -20 k

Example 4: A woman walks 30 m, W; then 40 m, N. Write her displacement in i, j notation and in R, q notation. In i, j notation, we have: +40 m R f -30 m R = R xi + R y j Rx = - 30 m Ry = + 40 m R = -30 i + 40 j Displacement is 30 m west and 40 m north of the starting position.

Example 4 (Cont. ): Next we find her displacement in R, q notation. +40 m R f -30 m q = 1800 – 53. 130 q = 126. 9 o R = 50 m (R, q) = (50 m, 126. 9 o)

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Question Magnitude =47. 2 q=122 degree

Converting Polar to Rectangular n r • y θ x 28