Use the definition of the scalar product, find the angles between the following pairs of vectors: A = 3i + 4j - 4k and B = 4i - 5j + 5k?

Solution:

Given A= 3i + 4j - 4k and B = 4i - 5j + 5k

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 = (3)(4) + (4)(-5) + (-4)(5)

A.B = 12 - 20 - 20

A⋅B = -28

Now, calculate the individual magnitude of A and B

|A| = √{3² + 4² + (-4)² }

∣A∣ = √41

∣B∣ = √{4² + (-5)² + 5²}

B = √66

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

A⋅B = |A| |B| cos(θ)

-28 = √41√66 cos(θ)

-28/√41√66 = cos(θ)

θ = cos^-1(-28 /√41√66)

θ ≈ 122.565°

The value of θ is 122.565°

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Use the definition of the scalar product, find the angles between the following pairs of vectors: A = 3i + 4j - 4k and B = 4i - 5j + 5k?

Summary:

By using the definition of the scalar product, the angle between the following pairs of vectors: A = 3i + 4j - 4k and B = 4i - 5j + 5k is 122.565°.