8.2 Vectors in the Coordinate Plane
Recall: Standard Position of a vector
A
B(6,8)
Vector AB can be described by the coordinates of its terminal point minus its initial point. This is called "Component Form".
Written as: <x2 x1 , y2 y1>
Example 1: Find the component form of AB with A(4, 2) and B(3, 5)
Example 2: Find the component form of XY with X(2, 7) and Y(6, 1)
Finding the magnitude (length) of a vector
= Use the Distance formula/Pythagorean Theorem!
Example 3: Find the magnitude of vector AB with A(4, 2) and B(3, 5)
Magnitude is denoted by |AB|
Vector Operations
Vectors can be added, subtracted, or multiplied by a scalar (constant).
a = <4, 1> b = <2, 5> c = <3, 0>
Find:
1) a + b
2) c 2b
3) 3b 2a + 3c
Unit Vector a vector with a magnitude of 1
**To find a unit vector, divide the vector by its magnitude.
Example 4: Find the component form, magnitude, and a unit vector for AB with A(5, 11) & B(3, 14)
Example 5: Find the unit vector for v = <4, 8>
Standard Unit Vectors
identity vectors denoted by the letters "i" and "j". Used to write vectors as linear combinations.
i = <1, 0>
j = <0, 1>
Example 6: Write DE as a linear combination of standard unit vectors with D(2, 3) & E(4, 5)
Finding vectors given Magnitude and Direction
Example 7: Find the component form of vector v with |v| = 10 and direction angle = 120 degrees
*SHORTCUT*
V = <|v|cosθ, |v|sinθ>
Example 8: Find the direction angle of vector p = 3i + 7j
*SHORTCUT*
θ = tan1( )yx
Example 9: Find the direction angle of r = <4, 5>