{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Engineering Calculus Notes 35

# Engineering Calculus Notes 35 - 1.2 VECTORS AND THEIR...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1.2. VECTORS AND THEIR ARITHMETIC 23 → − v +→ − w century; the ﬁrst elementary exposition was given by Edwin Bidwell Wilson (1879-1964) in 1901 [54], based on lectures by the American mathematical physicist Josiah Willard Gibbs (1839-1903)3 [17]. By a geometric vector in R3 (or R2 ) we will mean an “arrow” which can be moved to any position, provided its direction and length are maintained.4 We will denote vectors with a letter surmounted by an arrow, → like this: − . We deﬁne two operations on vectors. The sum of two vectors v → is formed by moving − so that its “tail” coincides in position with the w → − , then forming the vector − + − whose tail coincides with →→ “head” of v v w → − and whose head coincides with that of − (Figure 1.18). If → that of v w → − w → − v Figure 1.18: Sum of two vectors → instead we place − with its tail at the position previously occupied by the w → − and then move − so that its tail coincides with the head of − , → → tail of v v w →→ − + − , and it is clear that these two conﬁgurations form a we form w v parallelogram with diagonal →→→→ − +− =− +− v w w v (Figure 5.18). This is the commutative property of vector addition. A second operation is scaling or multiplication of a vector by a number. We naturally deﬁne →→ 1− = − v v →→→ 2− = − + − v v v →→→→ − = − + − + − = 2− + − →→ 3v v v v v v and so on, and then deﬁne rational multiples by → → → → − = m − ⇔ n − = m− ; w v w v n 3 I learned much of this from Sandro Caparrini [6, 7, 8]. This narrative diﬀers from the standard one, given by Michael Crowe [10] 4 This mobility is sometimes expressed by saying it is a free vector. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online