Calculate (A⃗ * B⃗) ⋅C⃗ for the three vectors A⃗ with magnitude A = 5.08 and angle θA = 25.6 ∘ measured in the sense from the + x - axis toward the + y - axis, B⃗ with B = 3.94 and θB = 63.9 ∘, and C⃗ with magnitude C = 6.16 and in the + z - direction. Vectors A⃗ and B⃗ are in the xy-plane. (A⃗ * B⃗) ⋅C⃗ (A → * B →) ⋅ C → = nothing

Question

Mathematics

Miya

24 April, 16:18

Calculate (A⃗ * B⃗) ⋅C⃗ for the three vectors A⃗ with magnitude A = 5.08 and angle θA = 25.6 ∘ measured in the sense from the + x - axis toward the + y - axis, B⃗ with B = 3.94 and θB = 63.9 ∘, and C⃗ with magnitude C = 6.16 and in the + z - direction. Vectors A⃗ and B⃗ are in the xy-plane. (A⃗ * B⃗) ⋅C⃗ (A → * B →) ⋅ C → = nothing

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Manuel Strong · Answer 1

(A⃗ * B⃗) ⋅C⃗ = - 76.415

Step-by-step explanation:

First we need to calculate (A⃗ * B⃗):

(A⃗ * B⃗) = A. B. sin (α). n

Where A is the magnitude of A⃗

Where B is the magnitude of B⃗

Where α is the angle between A⃗ and B⃗ = 63.9 - 25.6 = 38.3

Finally n is the vector orthogonal to A⃗ and B⃗

n magnitude is 1 and his direction is given by the right hand-rule

so n = (0, 0, 1)

(A⃗ * B⃗) = A. B. sin (α). n = 5.08. 3.94. sin (38.3). (0, 0, 1) = (0,0,12.4)

C⃗ can be written as C. (0,0,-1) because of his + z - direction

C. (0,0,-1) = 6.16. (0,0,-1) = (0,0,-6.16)

(A⃗ * B⃗) ⋅C⃗ = (0,0,12.4). (0,0,-6.16) = - 76.41480787 = - 76.415