The Dot Product

Sections 6.7

Objectives• Calculate the dot product of two

vectors. • Calculate the angle between two

vectors. • Use the dot product to determine

if two vectors are orthogonal, parallel, or neither.

Vocabulary• dot product

• orthogonal

• parallel

a product of two vectors formed by summing the product of their vertical components and the product of their horizontal components which produces a scalar number (not a vector)

two vectors are orthogonal if their dot product is 0

two vectors are parallel if the angle between them is either 0° or 180°

Formulas• Dot Product of and

OR

• Angle between two vectors (θ is the smallest non-negative angle between the two vectors)

jiv 11 ba jiw 22 ba

wvwv1cos

2121 bbaa wv

wvwv

cos

coswvwv

and

Compute the dot product of each pair of vectors:

)01()01( jiji

jj

)10()01( jiji

continued on next slide

Our first step in find each of these dot products is to put the vectors into standard rectangular coordinates. The vector i in rectangular coordinates is i = 1i + 0j. The vector j in rectangular coordinates is j = 0i + 1j.

is

)10()10( jiji

ji

ii

is

is

Compute the dot product of each pair of vectors:

ii2121 bbaa wv

continued on next slide

)01()01( jiji is

For this problem, the dot product formula to use is

In our problem a1 is 1, a2 is 1, b1 is 0, and b2 is 0.

1

)0)(0()1)(1(

ii

ii

Compute the dot product of each pair of vectors:

ji

continued on next slide

2121 bbaa wv

)10()01( jiji is

For this problem, the dot product formula to use is

In our problem a1 is 1, a2 is 0, b1 is 0, and b2 is 1.

0

)1)(0()0)(1(

ji

ji

Compute the dot product of each pair of vectors:

jj 2121 bbaa wv

)10()10( jiji is

For this problem, the dot product formula to use is

In our problem a1 is 0, a2 is 0, b1 is 1, and b2 is 1.

1

)1)(1()0)(0(

jj

jj

Given the vectors u = 8i + 8j and v = —10i + 11j find the following.

vu2121 bbaa wv

)1110()88( jiji is

For this problem, the dot product formula to use is

In our problem a1 is 8, a2 is -10, b1 is 8, and b2 is 11.

8

8880

)11)(8()10)(8(

vu

vu

vu

continued on next slide

Given the vectors u = 8i + 8j and v = —10i + 11j find the following.

uv2121 bbaa wv

)88()1110( jiji is

For this problem, the dot product formula to use is

In our problem a1 is -10, a2 is 8, b1 is 11, and b2 is 8.

8

8880

)8)(11()8)(10(

uv

uv

uv

continued on next slide

You should notice that when we computed the dot product with the vectors u•v first we got the same answer as when we switched the order of the vectors and calculated the dot product as v•u. This means that the dot product is commutative.

Given the vectors u = 8i + 8j and v = —10i + 11j find the following.

vv2121 bbaa wv

)1110()1110( jiji is

For this problem, the dot product formula to use is

In our problem a1 is -10, a2 is -10, b1 is 11, and b2 is 11.

221

121100

)11)(11()10)(10(

vv

vv

vv

Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.

wvu

continued on next slide

2121 bbaa wv

jiwv

jjiiwv

jijiwv

jijiwv

181

711910

791110

)79()1110(

We start by doing the part in the parentheses

For this problem, the dot product formula to use is

Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.

wvu

continued on next slide

2121 bbaa wv

We will now do the dot product of our result from the previous slide and the vector u.

For this problem, the dot product formula to use is

In our problem a1 is 8, a2 is -1, b1 is 8, and b2 is 18.

136)(

1448)(

)18)(8()1)(8()(

)181()88()(

wvu

wvu

wvu

jijiwvu

Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.

wuvu

continued on next slide

2121 bbaa wv

We start with doing the dot product of u and v and the dot product of u and w.

For this problem, the dot product formula to use is

8

8880

)11)(8()10)(8(

)1110()88(

vu

vu

vu

jijivu

128

5672

)7)(8()9)(8(

)79()88(

wu

wu

wu

jijiwu

Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.

wuvu

continued on next slide

136

1288

wuvu

wuvu

Now we will add the dot products from the previous slide.

8 vu 128 wu

Notice that this is the same as the answer that we got for u•(v + w). This tells us that the dot product can be distributed over addition and subtraction.

Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following. wwuv 123

continued on next slide

2121 bbaa wv

We start with doing the dot product of v and u and the dot product of w and w.

For this problem, the dot product formula to use is

8

8880

)8)(11()8)(10(

)88()1110(

uv

uv

uv

jijiuv

130

4981

)7)(7()9)(9(

)79()79(

wu

ww

ww

jijiww

Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following. wwuv 123

1536)(12)(3

156024)(12)(3

)130(12)8(3)(12)(3

wwuv

wwuv

wwuv

Now we multiply by the dot products we found by 3 and 12, respectively, and subtract the two results.

Find the angle θ in degrees measured between the vectors u = 10i + 3j and v = 1i — 7j .For this problem, we want to use the formula for the angle

between two vectors.

For this we need the to find the magnitude of each vector and the dot product of the two vectors.

wvwv1cos

2121 bbaa wvFor this problem, the dot product formula to use is

continued on next slide

In our problem a1 is 10, a2 is 1, b1 is 3, and b2 is -7.

11

2110

)7)(3()1)(10(

vu

vu

vu

Find the angle θ in degrees measured between the vectors u = 10i + 3j and v = 1i — 7j .

wvwv1cos

Now that we have the dot product, we need to find the magnitude of each vector. We can use the alternate formula for magnitude.

109

9100

)3()10( 22

u

u

u

Magnitude of u

continued on next slide

11 vu

22 ba v alternate magnitude formula.

Magnitude of v

50

491

)7()1( 22

v

v

v

Find the angle θ in degrees measured between the vectors u = 10i + 3j and v = 1i — 7j .

wvwv1cos

Now we are set to plug everything into the formula to find the angle between the vectors u and v.

109u 11 vu

56914188.98

5450

11cos

50109

11cos

cos

1

1

1

vuvu

50v

Determine if the pair of vectors is orthogonal, parallel, or neither.

jivjiu316

2and38

continued on next slide

The first step necessary to answer this question is to find the dot product of the two vectors. If the dot product is 0, then the vectors are orthogonal and we can stop. If the dot product is not 0, then we must go on to find the angle between the vectors.

2121 bbaa wvFor this problem, the dot product formula to use is

In our problem a1 is -8, a2 is 2, b1 is 3, and b2 is 16/3.

0

1616316

)3()2)(8(

316

2)38(

vu

vu

vu

jijivu

Since the dot product is 0, we can say that the vectors u and v are orthogonal.

Determine if the pair of vectors is orthogonal, parallel, or neither.

jivjiu 921and37

continued on next slide

2121 bbaa wvFor this problem, the dot product formula to use is

In our problem a1 is 7, a2 is 21, b1 is -3, and b2 is -9.

174

27147

)9)(3()21)(7(

316

2)38(

vu

vu

vu

jijivu

Since the dot product is not equal to 0, we know that the two vectors are not orthogonal. This means we need to continue with the formula for finding the angle between the two vectors by calculating the magnitude of each vector.

Determine if the pair of vectors is orthogonal, parallel, or neither.

jivjiu 921and37

continued on next slide

wvwv1cos

Now that we have the dot product, we need to find the magnitude of each vector. We can use the alternate formula for magnitude.

58

949

)3()7( 22

u

u

u

Magnitude of u

174 vu

22 ba v alternate magnitude formula.

Magnitude of v

522

81441

)9()21( 22

v

v

v

Determine if the pair of vectors is orthogonal, parallel, or neither. jivjiu 921and37

wvwv1cos

Now we are set to plug everything into the formula to find the angle between the vectors u and v.

58u 174 vu

01cos

174174

cos

30276

174cos

52258

174cos

cos

1

1

1

1

1

vuvu

522v

Since the angle between the two vectors is 0 degrees, the vectors are parallel.