Examples of Vector Quantities Scalar Multiple of a

Examples of Vector Quantities

Scalar Multiple of a Vector Parallel : Nonzero scalar multiple of each other (if & only if)

Sum of Vectors

Difference of Vectors

Vectors in a Coordinate Plane Position Vector Analytic Geometry : Connect Algebra & Geometry

Vector is Position Free

Sum & Difference of Position Vectors

Definition a = (a 1, a 2) , b = (b 1, b 2) 1) Addition a + b = (a 1+b 1, a 2+b 2) 2) Scalar Multiplication ka = (ka 1, ka 2) 3) Equality a = b if & only if a 1 = b 1, a 2 = b 2

Properties of Vectors 1. 2. 3. 4. 5. 6. 7. 8. 9. a+b=b+a (Commutative Law) a + (b + c ) = (a + b) + c (Associative Law) a+0=a (Additive Identity) a + (-a) = 0 (Additive Inverse) k(a+b) = ka + kb k : scalar (k 1 + k 2)a = k 1 a + k 2 a k 1(k 2 a) = (k 1 k 2)a 1 a = a 0 a = 0 (Zero Vector)

Magnitude(Length, Norm) of Vectors a(a 1, a 2) u Unit Vector u = (1/||a||)a

i, j : Basis of R 2 a(a 1, a 2) = a 1 i + a 2 j a 1 : Horizontal Component a 2 : Vertical Component

7. 1 Vectors in 2 -Space Figure 7. 11

Cartesian Coordinates

Position Vector

Vector Addition

i, j, k : Basis of R 3 a(a 1, a 2, a 3) = a 1 i + a 2 j + a 3 k

Definition a = (a 1, a 2, a 3) , b = (b 1, b 2, b 3) 1) Addition a + b = (a 1+b 1, a 2+b 2, a 3+b 3) 2) Scalar Multiplication ka = (ka 1, ka 2, ka 3) 3) Equality a = b if & only if a 1 = b 1, a 2 = b 2, a 3=b 3 4) Negative -b = (-b 1, -b 2, -b 3) 5) Subtraction a-b = a + (-b) = (a 1 -b 1, a 2 -b 2, a 3 -b 3) 6) Zero Vector 0 =(0, 0, 0) 7) Magnitude

7. 2 Vectors in 3 -Space Figure 7. 30