Chapter 3 Vectors in n-space Norm, Dot Product, and Distance in n-space Orthogonality.
The Dot Product Sections 6.7. Objectives Calculate the dot product of two vectors. Calculate the...
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Transcript of The Dot Product Sections 6.7. Objectives Calculate the dot product of two vectors. Calculate the...
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.