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7.2 The Dot Product

This document provides an overview of the dot product of vectors. It defines the dot product as the product of the magnitudes of two vectors and the cosine of the angle between them. It then lists several properties of the dot product, including commutativity, distributivity, and using Cartesian coordinates. Examples are provided to demonstrate calculating the dot product, using its properties to simplify expressions, determining if vectors are collinear or orthogonal, and finding a vector orthogonal to a given vector. Homework problems are assigned from the textbook.

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Gary Hoang
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241 views4 pages

7.2 The Dot Product

This document provides an overview of the dot product of vectors. It defines the dot product as the product of the magnitudes of two vectors and the cosine of the angle between them. It then lists several properties of the dot product, including commutativity, distributivity, and using Cartesian coordinates. Examples are provided to demonstrate calculating the dot product, using its properties to simplify expressions, determining if vectors are collinear or orthogonal, and finding a vector orthogonal to a given vector. Homework problems are assigned from the textbook.

Uploaded by

Gary Hoang
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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CALCULUS AND VECTORS (MCV4U)

UNIT 4: GEOMETRIC AND ALGEBRAIC VECTORS

7.2 The Dot Product There are two types of vector multiplication: (1) the dot product or scalar product, which results in a scalar; (2) the cross product or vector product, which results in a vector. 1. The Definition of the Dot Product The dot product of two vectors u and v , denoted by u v , is defined as u v = u v cos (0 180)

2. Properties of the Dot Product (1) Commutative Law: u v = v u

(2) Distributive Law: u (v + w ) = u v + u w (3) k( u v ) = (ku) v = u (kv ) = ( u v ) k, k (Associative Law?)


2 (4) u u = u

j j (5) i i = = k k = 1

(6) For non-zero vectors u and v , u v = 0 u and v are perpendicular.

j j 0 (7) i = i =

k = k = 0 j j
k i = i k = 0

3. The Dot Product for Cartesian Vectors We can also use the coordinates to work out the dot product, without having to know the angle between the vectors.

If u = u1i + u2 + u3 k and v = v1i + v 2 + v 3 k , then j j


u v = (u1i + u2 + u3 k ) (v1i + v 2 + v 3 k ) j j j = u1v1 (i i ) + u1v 2 (i ) + u1v 3 (i k ) + u2v1 ( i ) + u2v 2 ( ) + u2v 3 ( k ) j j j j j + u3v1 ( k i ) + u3v 2 ( k ) + u3v 3 ( k k ) = u1v1 + u2v 2 + u3v 3

Example 1:

a. Find the dot product of u b. Find the dot product of u c. Find the angle between u Solution: a. u v = (24)(9)cos 34
179.1
b. u v = (1, 0) (2 8) = (1)(2) + (0)(8) = 2

and v if u = 24 , v = 9 , and = 34. and v if u = (1, 0) and v = (2, 8) . and v if u = (3, 1, 2) and v = (5, 4, 1) .

u v c. cos = uv = = (3)(5) + (1)(4) + (2)(1) (3) 2 +12 + 2 2 5 2 + (4) 2 + (1) 2

3 2 = 150

Example 2: Use the properties of the dot product to expand and simplify each expression. a. (ku) ( u + v ) b. ( r + s ) ( r s )

Solution:
a. (ku) ( u + v ) = (ku) u + (ku) v = k( u u) + k( u v ) 2 = k u + k( u v )

b. ( r + s ) ( r s ) = ( r + s ) r + ( r + s ) (s ) = r r + s r r s s s 2 2 = r + r s r s s 2 2 =r s

Example 3:

Determine if the vectors a = (6, 2, 4) and b = (9, 3, 6) are collinear using the dot product.
Solution: a b cos = ab

(6)(9) + (2)(3) + (4)(6) 6 2 + 2 2 + 4 2 9 2 + 32 + 6 2

Therefore a and b are collinear.

=1 = 0

Example 4:

a. Prove that two non-zero vectors u and v are orthogonal (perpendicular) if and only if u v = 0 .
b. Find a vector that is orthogonal to (3, 4, 5). Solution:

a. First, prove that if u and v are orthogonal, then u v = 0 . If u and v are non-zero and orthogonal, then the angle between them is 90. Thus, u v = u v cos90 = 0

Next, prove that if u v = 0 , then u and v are orthogonal. Let u v = 0 , then u v = u v cos = 0 But u 0, v 0 , thus
cos = 0 = 90

b. Let u = (x, y, z) be the vector that is orthogonal to (3, 4, 5), then

(3, 4, 5) (x, y, z) = 0 3x + 4 y + 5z = 0

This equation has infinitely many solution sets for (x, y, z) , that is, there are infinitely many vectors orthogonal to (3, 4, 5).
To find a particular vector solution, choose any values (not both zero) for two of the variables and 4 4 solve for the third variable. E.g. let x = 0, y = 1, then z = . Therefore u = 0,1, . 5 5

Homework: Pg. 375: C1-C4 Pg. 375: 1, 2c)-f), 3, 4c)-f), 5-8, 12 Pg. 399: C5 Pg. 400: 15 f)-i), 16, 17, 23-33

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