VIEWS: 14 PAGES: 17 POSTED ON: 12/10/2011
Vectors Scalars & Vectors • Vectors – Quantity with both magnitude & direction – Does NOT follow Head elementary arithmetic/algebra rules Direction/Angle – Examples – position, Tail force, moment, Line of Action velocities, acceleration Parallelogram Law • The resultant of two forces can be obtained by – Joining the vectors at their tails A A+B Constructing a B parallelogram 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 B A YES! - commutative Vector Subtraction A-B = A + (-B) A -B B -B R A Vector Subtraction • Does A–B = B-A ? -B B R A -R -A NO! – opposite sense 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 A( x1 , y1 , z1 ) and B( x2 , y2 , z2 ), the vector a with representation AB is a x2 x1, y2 y1, z2 z1 Find the vector represented by the directed line segment with initial point A(2,-3,4) and terminal point B(-2,1,1). a 2 2,1 (3),1 4 a 4, 4, 3 Magnitude of a vector Determine the magnitude of the following: Example If a 4,0,3 and b 2,1,5 , find a and the vectors a+b, a-b, 3b,and 2a+5b. a b 4, 0,3 2,1,5 a 4 0 3 25 5 2 2 2 a b 4 2, 0 1,3 5 a b 2,1,8 3b 3 2,1,5 3(2),3(1),3(5) 6,3,15 a b 4, 0,3 2,1,5 a b 4 (2), 0 1,3 5 2a 5b 2 4, 0,3 5 2,1,5 a b 6, 1, 2 2a 5b 8, 0, 6 10,5, 25 2a 5b 2,5,31 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 These are not parallel b= -3a 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? i= 1,0,0 j= 0,1,0 k= 0,0,1 Standard Basis Vectors If a= a1 , a2 , a3 , then we can write a= a1 , a2 , a3 a1 , 0, 0 0, a2 , 0 0, 0, a3 a=a1 1,0,0 a2 0,1,0 a3 0,0,1 a=a1i a2 j a3k Example- Write 1, 2,6 in terms of the standard basis vector i,j,k. 1, 2,6 i - 2j 6k 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 1 a The unit vector in the same direction of a is u= a= a a 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 2i - j - 2k 22 (1)2 (2)2 9 3 1 a u= a= Check? a a Same direction? 1 u= a Magnitude = 1? 3 1 u= (2i - j - 2k) 3 2 1 2 u= i - j - k 3 3 3 Homework • P649 – 4,5,7,9,11,15,17,19