Multivariate Calculus - Bruce E. Shapiro
Multivariate Calculus - Bruce E. Shapiro Multivariate Calculus - Bruce E. Shapiro
12 LECTURE 2. VECTORS IN 3D To obtain u - v, start (1) by placing the second vector at the head of the first and then (2) reflect it across its own tail to find −u. Finally, (3) draw the arrow from the tail of u to the head of the reflection of v. Observe that this process is not reversible: u - v is not the same as v - u. In terms of components, u − v = (u x , u y , u z ) − (v x , v y , v z ) = (u x − v x , u y − v y , u z − v z ) = −(v − u) (2.13) Figure 2.4: Vector subtraction, demonstrating construction of v −u (left) and v −u (right). Observe that v − u = −(v − u) Theorem 2.3 Properties of Vector Addition & Scalar Multiplication. Let v, u, and w be vectors, and a, b ∈ R be real numbers. Then the following properties hold: 1. Vector addition commutes: 2. Vector addition is associative: v + u = u + v (2.14) (u + v) + w = u + (v + w) (2.15) 3. Scalar multiplication is distributive across vector addition: 4. Identity for scalar multiplication 5. Properties of the zero vector (a + b)v = av + bv (2.16) a(v + w) = av + aw (2.17) 1v = v1 = v (2.18) 0v = 0 (2.19) 0 + v = v + 0 = v (2.20) Revised December 6, 2006. Math 250, Fall 2006
LECTURE 2. VECTORS IN 3D 13 Definition 2.7 The dot product or scalar product u · v between two vectors v = (v x , v y , v z ) and u = (u x , u y , u z ) is the scalar (number) u · v = u x v x + u y v y + u z v z (2.21) Sometimes we will find it convenient to represent vectors as column matrices. The column matrix representation of v is given by the components v x , v y , and v z represented in a column, e.g., ⎛ ⎞ ⎛ ⎞ v x u x v = ⎝v y ⎠ , u = ⎝u y ⎠ (2.22) vz uz With this notation, the transpose of a vector is u T = ( u x u y u z ) (2.23) and the dot product is ⎛ ⎞ u · v = u T v = ( ) v x u x u y u z ⎝v y ⎠ = u x v x + u y v y + u z v z (2.24) v z which gives the same result as equation (2.21). Example 2.1 Find the dot product of ⃗u = (1, −3, 7) and ⃗v = (16, 4, 1) Solution. By equation (2.21), u · v = (1)(16) + (−3)(4) + (7)(1) = 16 − 12 + 7 = 11. Theorem 2.4 ‖v‖ = √ v · v Proof. From equation (2.21), v · v = v x v x + v y v y + v z v z = ‖v‖ 2 where the last equality follows from equation (2.6). Example 2.2 Find the lengths of the vectors ⃗u = (1, −3, 7) and ⃗v = (16, 4, 1) Solutions. ‖u‖ = √ (1) 2 + (−3) 2 + (7) 2 = √ 1 + 9 + 49 = √ 59 ≈ 7.681 ‖v‖ = √ (16) 2 + (4) 2 + (1) 2 = √ 256 + 16 + 1 = √ 273 ≈ 16.523 Math 250, Fall 2006 Revised December 6, 2006.
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LECTURE 2. VECTORS IN 3D 13<br />
Definition 2.7 The dot product or scalar product u · v between two vectors<br />
v = (v x , v y , v z ) and u = (u x , u y , u z ) is the scalar (number)<br />
u · v = u x v x + u y v y + u z v z (2.21)<br />
Sometimes we will find it convenient to represent vectors as column matrices.<br />
The column matrix representation of v is given by the components v x , v y , and v z<br />
represented in a column, e.g.,<br />
⎛ ⎞ ⎛ ⎞<br />
v x<br />
u x<br />
v = ⎝v y<br />
⎠ , u = ⎝u y<br />
⎠ (2.22)<br />
vz uz<br />
With this notation, the transpose of a vector is<br />
u T = ( u x u y u z<br />
)<br />
(2.23)<br />
and the dot product is<br />
⎛ ⎞<br />
u · v = u T v = ( )<br />
v x<br />
u x u y u z<br />
⎝v y<br />
⎠ = u x v x + u y v y + u z v z (2.24)<br />
v z<br />
which gives the same result as equation (2.21).<br />
Example 2.1 Find the dot product of ⃗u = (1, −3, 7) and ⃗v = (16, 4, 1)<br />
Solution. By equation (2.21),<br />
u · v = (1)(16) + (−3)(4) + (7)(1) = 16 − 12 + 7 = 11.<br />
Theorem 2.4 ‖v‖ = √ v · v<br />
Proof. From equation (2.21),<br />
v · v = v x v x + v y v y + v z v z = ‖v‖ 2<br />
where the last equality follows from equation (2.6).<br />
Example 2.2 Find the lengths of the vectors ⃗u = (1, −3, 7) and ⃗v = (16, 4, 1)<br />
Solutions.<br />
‖u‖ = √ (1) 2 + (−3) 2 + (7) 2 = √ 1 + 9 + 49 = √ 59 ≈ 7.681<br />
‖v‖ = √ (16) 2 + (4) 2 + (1) 2 = √ 256 + 16 + 1 = √ 273 ≈ 16.523<br />
Math 250, Fall 2006 Revised December 6, 2006.
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