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# vector - VECTORS DOT PRODUCT O Knill Math21a HOMEWORK...

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VECTORS, DOT PRODUCT O. Knill, Math21a HOMEWORK: Section 11.1: 34,36, Section 11.2: 76, Section 11.3: 20, 62, Due Friday, September 29 VECTORS. Two points P = ( x 1 , y 1 ), Q = ( x 2 , y 2 ) in the plane define a vector vectorv = ( x 2 x 1 , y 2 y 1 ) . It points from P to Q and we can write P + vectorv = Q . COMPONENTS. Points P in space are in one to one correspondence to vectors pointing from 0 to P . The numbers vectorv i in a vector vectorv = ( v 1 , v 2 ) are also called components or of the vector. REMARKS: vectors can be drawn everywhere in the plane. If a vector starts at the origin O , then the vector vectorv = ( v 1 , v 2 ) points to the point ( v 1 , v 2 ) . That’s is why one can identify points P = ( a, b ) with vectors vectorv = ( a, b ) . Two vectors which can be translated into each other are considered equal . They have the same components. ADDITION, SUBTRACTION AND SCALAR MULTIPLICATION. x y u v u+v x y u v u-v x y u 3 u vectoru + vectorv = ( u 1 , u 2 ) + ( v 1 , v 2 ) = ( u 1 + v 1 , u 2 + v 2 ) vectoru vectorv = ( u 1 , u 2 ) − ( v 1 , v 2 ) = ( u 1 v 1 , u 2 v 2 ) λvectoru = λ ( u 1 , u 2 ) = ( λu 1 , λu 2 ) BASIS VECTORS. The vectors vector i = ( 1 , 0 ) , vector j = ( 0 , 1 ) are called standard basis vectors in the plane. In space, one has the basis vectors vector i = ( 1 , 0 , 0 ) , vector j = ( 0 , 1 , 0 ) , vector k = ( 0 , 0 , 1 ) . Every vector vectorv = ( v 1 , v 2 ) in the plane can be written as vectorv = v 1 vector i + v 2 vector j . Every vector vectorv = ( v 1 , v 2 , v 3 ) in space can be written as vectorv = v 1 vector i + v 2 vector j + v 3 vector k .
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