1. Vectors

2. Scalars & Vectors Vectors Quantity with both magnitude & direction
Does NOT follow elementary arithmetic/algebra rules
Examples – position, force, moment, velocities, acceleration
Magnitude
Head
Direction/Angle
Tail
Line of Action

3. Parallelogram Law The resultant of two forces can be obtained by
Joining the vectors at their tails A
A+B
Constructing a
parallelogram B
The resultant is the diagonal of the parallelogram

4. Triangle Construction
The resultant of two forces can be obtained by
Joining the vectors in tip-to-tail fashion A B R
The resultant extends
from the tail of A to
the head of the B

5. Vector Addition
Does
A + B = B + A ? A B R R A B
YES! - commutative

6. Vector Subtraction
A - B = A + (-B) A -B B -B R A

7. Vector Subtraction Does A – B = B - A ? NO! – opposite sense -B B R -R

8. Vector Operations
Multiplication & Division of Vector (A) by Scalar (a)
a * A = aA 2A
2 * A = 2A A
-.5 * A = -.5A A
-.5A

9. Representation of a Vector
Given the points and , the vector a with representation is a
Find the vector represented by the directed line segment with initial point A(2,-3,4) and terminal point B(-2,1,1).

10. Magnitude of a vector
Determine the magnitude of the following:

11. Example

12. Parallel These are parallel since
Two vectors are parallel to each other if one is the scalar multiple of the other.
Determine if the two vectors are parallel
These are parallel since
b= -3a
These are not parallel since 4(1/2) =2 , but 10(1/2)=5 not -9

13. Unit vectors
Any vector that has a magnitude of 1 is considered a unit vector.
Can you think of a unit vector?

14. Standard Basis Vectors
Example- Write in terms of the standard basis vector i,j,k.

15. Example
If a = i + 2j - 3k and b = 4i + 7k, express the vector 2a+3b in terms of i,j,k.
2a+3b=2(i + 2j - 3k)+3(4i + 7k)
2a+3b=2i + 4j - 6k+ 12i + 21k
2a+3b=14i+4j+15k

16. Unit Vectors The unit vector in the same direction of a is
Find a unit vector in the same direction as 2i – j – 2k.
We are looking for a vector in the same direction as the original vector, but is also a unit vector.
Let’s first find the magnitude
Check?
Same direction?
Magnitude = 1?

17. Homework
P649
4,5,7,9,11,15,17,19