Need help with these questions. #12,13,28,52,53 only Which property of dot produ

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#12,13,28,52,53 only

Which property of dot products allows us to conclude that if v is orthogonal to both u and w, then v is orthogonal to u + w? Which is the projection of v along v: (a) v or (b ) ev? Exercises In Exercises 1-12, compute the dot product. 1, 2, 1 middot 4. 3, 5 3, -2,. 2 middot 1, 0, 1 0, 1, 0 middot 7, 41, -3 1, 1, 1 middot 6, 4, 2 3, 1 middot 4. -7 k middot J k middot k (1 + J) middot (J + k) (3J + 2k) middot (1 - 4k) (i + j + k) middot (3i + 2j + 5k) (-k ) middot (1 - 2J + 7k) In Exercises 13-18, determine whether the two vectors are orthogonal if not, whether the angle between them is acute or obtuse. 1, 1, 1, 1, -2, -2 0, 2, 4, -5, 0, 0 k middot J k middot k (1 + J) middot (J + k) (3J + 2k) middot (1 - 4k) (i + j + k) middot (3i + 2j + 5k) (-k ) middot (1 - 2J + 7k) In Exercises 13-18, determine whether the two vectors are orthogonal if not, whether the angle between them is acute or obtuse. 1, 1, 1, 1, -2, -2 0, 2, 4, -5, 0, 0 1, 2, 1 middot 7, -3, -1 0, 2, 4 middot 3, 1, 0 12, 6 middot 2, -4 In Exercises 19-22 find the consine of the angle between the vectors 0, 3, 1 , 4, 0, 0 1, 1, 1 , 2, -1, 2 i + j, j + 2k 3i + k, i+ j + k In Exercises 23-28 find the angle between the vectors. Use a calculator if necessary 1, 1, 1 , 1, 0, 1 3, 1, 1 , 2, -4, 2 0, 1, 1 , 1, -1, 0 1, 1, -1 , 1, -2, -1 Find all values of b for which the vectors are orthogonal. b, 3, 2 , 1, b, 1 4, -2, 7 , b2, b, 0 Find a vector that is orthogonal

Explanation / Answer

12)

(-k).(i-2j+7k)

=(0+0+(7*-1)) = -7

13)

(1,1,1) and (1,-2,-2)

A.B = |A||B| cos(theta)

1-2-2 = sqrt(3)*sqrt(9)*cos(theta)

cos(theta) = -3/3*sqrt(3)

cos(theta) = -1/sqrt(3)

theta = arccos(-1/sqrt(3)) = 125.26

so, the angle is obtuse

28)

(1,1,-1) and (1,-2,-1)

A.B = |A||B| cos(theta)

1-2+1 = sqrt(3)*sqrt(6) *cos(theta)

cos(theta) = 0

so theta = 90 deg

52)

the projection of U onto V is:

(U dot V)/|V| = (2-6)/sqrt(1+4) = -4/sqrt(5) = -1.79

53)

the projection of U onto V is:

(U dot V)/|V| = -2/sqrt(5) = -0.89

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