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Lecture 2

CSE 167 Lecture 2: L2 1/10/19Premium

3 pages105 viewsWinter 2019

Department
Computer Science and Engineering
Course Code
CSE 167
Professor
Ravi Ramamoorthi
Lecture
2

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Vector Addition
Calculated using the parallelogram rule
Commutative! a + b = b + a
Cartesian Coordinates
X and Y can be any (usually orthogonal unit) vectors
Sum vectors using their X and Y coordinates (Xa + Xb, Ya + Yb)
Vector Multiplication
Use RHR (right hand rule) to associate the X, Y, and Z axes
Dot (scalar) product
a b = ||a|| ||b|| cosθ
|b a| = ||b|| cosθ = a b / ||a||
- “Length of vector b ALONG the direction of a” (aka projection of b onto a)
As a unit vector:
b a = ||b a|| x (a / ||a||) = (a b) a / ||a||2
Note:
a • (b + c) = a b + a c
Dot Product in Cartesian Components
a b = xaxb + yayb (Why?)
a = xa x + ya y
B = xb x + yb y
a b = (xa x + ya y) • (xb x + yb y)
= (xaxb) xx + (yayb) yy + (xayb) xy + (yaxb) yx
Note:
xx = 1 (same for y)
xy = 0 (same for yx)
= xaxb + yayb
Cross (vector) product
a x b = - b x a (order in which you do CP matters!)
||a x b|| = ||a|| ||b|| sinɸ
Cross product is orthogonal to the 2 initial vectors
Properties:
a x a = 0
a x (b + c) = a x b + a x c
a x (kb) = k(a x b)
a x b = A*b (where A is the “dual”matrix of vector a)
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