Vectors

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

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

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

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

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

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

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

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).

Magnitude of a vector Determine the magnitude of the following:

Example

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

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

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

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

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?

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