Vectors Scalars Vectors Vectors Quantity with both magnitude

Vectors

Scalars & Vectors • Vectors – Quantity with both magnitude & direction – Does NOT follow elementary arithmetic/algebra rules – Examples – position, force, moment, velocities, acceleration e ud t i n ag Head M Tail Direction/Angle 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 n The resultant extends from the tail of A to the head of the B

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

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

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

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

Representation of a Vector Given the points and the vector a with representation a , is 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 • 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= -3 a 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 + 2 j - 3 k and b = 4 i + 7 k, express the vector 2 a+3 b in terms of i, j, k. 2 a+3 b=2(i + 2 j - 3 k)+3(4 i + 7 k) 2 a+3 b=2 i + 4 j - 6 k+ 12 i + 21 k 2 a+3 b=14 i+4 j+15 k

Unit Vectors The unit vector in the same direction of a is Find a unit vector in the same direction as 2 i – j – 2 k. 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 • P 649 – 4, 5, 7, 9, 11, 15, 17, 19