6.3 Vectors
Mar 17­8:04 AM
Vector: has magnitude (length) and direction
terminal point
Q (x2, y2)
PQ
||v||
magnitude
2
2
√(x2 ­ x1) + (y2 ­ y1) P (x1, y1)
initial point
Mar 17­8:07 AM
1
Component Form of a Vector A vector whose initial point is at the origin (0, 0) can be represented by the coordinates of its terminal point (v1, v2). v = < v1 , v2 >
The component form of the vector with initial point P = (x1, y1) and terminal point Q = (x2, y2) is:
v = < v1 , v2 > = (x2 ­ x1 , y2 ­ y1)
The magnitude (or length) of v is:
||v|| = √(x2 ­ x1)2 + (y2 ­ y1)2 = √ v12 + v22
Mar 17­8:13 AM
Example 1: Find the component form and magnitude of the vector v that has (1, 7) as its initial point and (4, 3) as its terminal point.
Mar 17­8:20 AM
2
Vector Operations Geometrically, the product of a vector v and a scalar k is . . . the vector that is k times as long as v
If k is positive, kv has the same direction as v, and if k is negative, kv has the opposite direction.
v
1/2v
­v
2v
Mar 17­8:23 AM
To add two vectors geometrically, position them so the initial point of one coincides with the terminal point of the other
This technique is called the parallelogram law for vector addition because the vector u + v, often called the resultant of vector addition, is . . . the diagonal of a parallelogram with u and v as sides u
v
v
u
Mar 17­8:24 AM
3
Let u = 〈u1, u2〉 and v = 〈v1, v2〉 be vectors and let k be a scalar u + v = < u1 + v1 , u2 + v2 > (vector addition)
ku = < ku1 , ku2 > (Scalar multiplication)
Mar 17­8:25 AM
Let u = 〈1, 6〉 and v = 〈− 4, 2〉. Sketch the operations geometrically. Then find:
(a) 3u
(b) u + v
Mar 17­8:27 AM
4
Unit Vector: has a magnitude of 1 To find a unit vector u that has the same direction as vector v . . . u = v ||v||
Mar 17­8:29 AM
Find a unit vector in the direction of v = 〈− 8, 6〉.
Mar 17­8:32 AM
5
Standard Unit Vectors:
i = <1, 0> j = <0, 1>
Linear Combination
Vector v = 〈v1, v2〉
can also be represented as: j = <0, 1>
v = v1i + v2j i = <1, 0> Mar 17­8:34 AM
Let v = 〈− 5, 3〉. Write v as a linear combination of the standard unit vectors i and j.
Mar 17­8:39 AM
6