Unformatted text preview: ntities Combining Vectors Vectors in the Plane Vectors in Space The Norm Scalar Multiplication Deﬁnition
We deﬁne the scalar multiplication of a vectors v by a scalar λ to
be the vector λv having magnitude λ v and having the same
direction as v if λ > 0, and the opposite direction as v if λ < 0. u
u λu −u Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vector Diﬀerence
Deﬁnition
The diﬀerence of two vectors v and w is the vector v − w deﬁned
by
v − w = v + (−v). u−v v u Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Vectors in the Plane
To work with vectors, it is convenient to place them in a
coordinate system.
Suppose v lies in a plane with a Cartesian coordinate system.
If P (x0 , y0 ) is the coordinate of the tail of v, and Q (x1 , y1 ) is
the coordinate of the head of v, then we call P the initial
point and Q the terminal point of v.
−
→
We write v = PQ .
The magnitude of v is the length of the line segment from
P (x0 , y0 ) to Q (x1 , y1 ):
v= (x1 − x0 )2 + (y1 − y0 )2 = (∆x )2 + (∆y )2 . Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space Vector Components Suppose v has initial point P (x0 , y0 ) and terminal point
Q (x1 , y1 ).
Set v1 = x1 − x0 and v2 = y1 − y0 .
Note that
v= 2
2
v1 + v2 . If the initial point P is the origin, then v = x1 , y1 , the
coordinates of the terminal point.
We call v1 and v2 the components of v.
If v = v1 , v2 and w = w1 , w2 , then v = w if and only if
v1 = w1 and v2 = w2 . The Norm Outline Vector Quantities Combining Vectors Vectors in the Plane Vectors in Space The Norm Examples Example
1 Let v be the vector having initial point P (3, 4) and terminal
point Q (5, −1). If u has initial point O (0, 0) and w has initial
point A(−2, 6), ﬁnd their respective terminal points if
v = u = w. 2 Find the magnitude of the vector v having initial point P (3, 4)
and terminal point Q (5, −1). Outlin...
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 Spring '14
 KenyonJ.Platt
 Linear Algebra, Algebra, Vectors, Cartesian Coordinate System, Euclidean geometry, Standard basis, vector algebra

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