Using the definition of the scalar product, find the angle between the following vectors. (Find the smallest angle.)
(a) A = 5i - 9j and B = 5i - 5j
(b) A = -8i + 5j and B = 3i - 4j + 2k
(c) A= -2i + 2j and B = 3j + 4k

Solution:

a) Given A = 5i - 9j and B = 5i - 5j

The scalar product as two definitions:

[1] AB = (a_x)(b_x)+(a_y)(b_y)

[2] A.B = ∣A∣∣B∣cos(θ) where θ is the angle between the vectors.

We will use [1] to calculate the scalar product:

A⋅B = (5)(5) + (-9)(-5)

A.B = 25 + 45

A⋅B = 70

Now, calculate the individual magnitude of A and B

|A| = √{5² + (-9)²}

∣A∣ = √25 + 81

|A| = √106

∣B∣ = √{(-5)² + 5²}

|B| = √25 + 25

|B| = √50

We will use [2] to find the value of θ

A⋅B = |A| |B| cos(θ)

70 = √106√50 cos(θ)

70/√106√50 = cos(θ)

θ = cos^-1(70 / √106√50)

θ ≈ 15.9°

The value of θ is 15.9°

b) Given A = -8i + 5j + 0k and B = 3i - 4j + 2k

The scalar product as two definitions:

[1] AB = (a_x)(b_x)+(a_y)(b_y)+(a_z)(b_z)

[2] A.B = ∣A∣∣B∣cos(θ) where θ is the angle between the vectors.

We will use [1] to calculate the scalar product:

A⋅B = (-8)(3) + (5)(-4) + (0)(2)

A.B = -24 - 20

A⋅B = -44

Now, calculate the individual magnitude of A and B

|A| = √{5² + (-8)²}

∣A∣ = √25 + 64

|A| = √89

∣B∣ = √{3² + (-4)² + 2²}

|B| = √9 + 16 + 4

|B| = √29

We will use [2] to find the value of θ

A⋅B = |A| |B| cos(θ)

-44 = √89√29 cos(θ)

-44/√89√29 = cos(θ)

θ = cos^-1(-44 / √89√29)

θ = 29.7°

The value of θ is 29.7°

c) Given A= -2i + 2j and B = 3j + 4k

The scalar product as two definitions:

[1] AB = (a_x)(b_x)+(a_y)(b_y)

[2] A.B = ∣A∣∣B∣cos(θ) where θ is the angle between the vectors.

We will use [1] to calculate the scalar product:

A⋅B = (-2)(3) + (2)(4)

A.B = -6 +8

A⋅B = 2

Now, calculate the individual magnitude of A and B

|A| = √{2² + (-2)²}

∣A∣ = √4 + 4

|A| = √8

∣B∣ = √{4² + 3²}

|B| = √16 + 9

|B| = √25 = 5

We will use [2] to find the value of θ

A⋅B = |A| |B| cos(θ)

2 = √8√25 cos(θ)

2/√8√25 = cos(θ)

θ = cos^-1(2/√8√25)

θ ≈ 81.8°

The value of θ is 81.8°

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Using the definition of the scalar product, find the angle between the following vectors. (Find the smallest angle.
a) A = 5i - 9j and B = 5i - 5j
(b) A = -8i + 5j and B = 3i - 4j + 2k
(c) A= -2i + 2j and B = 3j + 4k

Summary:

Using the definition of the scalar product, the angle between the following vectors.

a) A = 5i - 9j and B = 5i - 5j is θ = 15.9°

(b) A = -8i + 5j and B = 3i - 4j + 2k is θ = 29.7°

(c) A= -2i + 2j and B = 3j + 4k is θ = 81.8°