SCI340_L05 _vectors.ppt More math concepts

Objectives Distinguish between vector and scalar quantities. Carry out addition, subtraction, and scalar multiplication of vectors.

What’s the Point? How can we specify quantities that depend on direction? How do such quantities combine?

Vectors and Scalars Vector: quantity needing a direction to fully specify (direction + magnitude) Scalar: directionless quantity Either can be united or unitless

Represent as Arrows direction: obvious magnitude: length location is irrelevant these are identical

Represent as Components Components: projections in (x, y) directions x y B A A = (4, 3) B = (0, –2)

Represent with Unit Vectors x = (1, 0, 0) y = (0, 1, 0) z = (0, 0, 1) Linear combination of unit vectors xx + yy + zz = (x, y, z)

Represent as polar coordinates Magnitude, angle Conventionally angle is ccw of +x axis A = 5, 36.87° A B = 2, 270° B

Polar ↔ Cartesian conversion (r, q) → (x, y) x = r cos(q) y = r sin(q) (x, y) ↔ (r, q) r2 = x2 + y2 tan(q) = y/x r y q x

Magnitude from Components Components: lengths of sides of right triangle Magnitude: length of hypotenuse A A = (4, 3) ||A ||= A = 42 + 32

Physics Vectors and Scalars Position, displacement, velocity, acceleration, and force are vector quantities. Mass and time are scalar quantities. (There are many others)

Add Vectors Head-to-tail A A B C B A + B = C

How to Add Vectors Graphically Place following vector’s tail at preceding vector’s head Resultant (vector sum) starts where the first vector starts and ends where the last vector ends Add any number of vectors, one after another

Adding by Components Resultant: Add (x, y) components individually C = A + B = (4+0, 3–2) = (4, 1)

Question Which vector is the sum of vectors A and B? a b B A c d

Group Work Draw two vectors A and B. Graphically find A + B.

Question Is vector addition commutative? Yes. No.

Vector Addition is Commutative B A + B = C B + A = C A + B = B + A

Respect the Units For a vector sum to be meaningful, the vectors you add must have the same units! Just as with scalars: good! 5 s + 10 s = 15 s 5 kg + 10 m = 15 ? Bad! Or, algebra in general: good! 5 a + 10 a = 15 a 5 b + 10 c = 15 ? Bad!

Subtract Vectors Add the negative of the vector being subtracted. (Negative = same magnitude, opposite direction: what you must add to get zero) D A B A –B –B A – B = A + (–B) = D

Group Work Make up three vectors A, B, and C. Graphically show: A – B A + B + C C + A + B

Multiplication by a Scalar Product of (scalar)(vector) is a vector The scalar multiplies the magnitude of the vector; direction does not change Direction reverses if scalar is negative 2 A –2 A A 1/2 A

Scalar Multiplication Example Velocity (a vector)  time (a scalar) v Dt = Dr Result is displacement (a vector). The vectors are in the same direction, but have different units!