1. SCI340_L05 _vectors.ppt More math concepts

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

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

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

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

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

7. 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)

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

9. 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

10. Magnitude from Components
Components: lengths of sides of right triangle
Magnitude: length of hypotenuse A
A = (4, 3)
||A ||= A =

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

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

13. 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

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

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

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

17. Question
Is vector addition commutative?
Yes.
No.

18. Vector Addition is CommutativeB
A + B = C
B + A = C
A + B = B + A

19. 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!

20. 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

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

22. 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

23. 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!