Final answer:
a) The dot product a ⋅ (b × c) is -36. b) The dot product a ⋅ (b + c) is -4. c) The cross product a × (b + c) is -6i + 6j + 4k.
Step-by-step explanation:
To find the dot product and cross product of the given vectors, we can use the formulas:
a) Dot product (scalar product): a ⋅ (b × c) = a ⋅ (b × c) = |a| |b| sinθ, where θ is the angle between the vectors.
Substituting the values, we get: a ⋅ (b × c) = (6i + 2j - 4k) ⋅ ((-2i - j + k) × (3i + 2j + 2k)).
Step 1: Calculate the cross product of b and c: (-2i - j + k) × (3i + 2j + 2k) = (-2 - 6)i + (4 - 6)j + (-6 + 2)k = -8i - 2j - 4k.
Step 2: Calculate the dot product with a: (6i + 2j - 4k) ⋅ (-8i - 2j - 4k) = (6)(-8) + (2)(-2) + (-4)(-4) = -48 - 4 + 16 = -36.
b) Dot product (scalar product): a ⋅ (b + c) = a ⋅ b + a ⋅ c = |a| |b| cosθ + |a| |c| cosθ, where θ is the angle between the vectors.
Substituting the values, we get: a ⋅ (b + c) = (6i + 2j - 4k) ⋅ ((-2i - j + k) + (3i + 2j + 2k)).
Step 1: Add b and c: (-2i - j + k) + (3i + 2j + 2k) = i + j + 3k.
Step 2: Calculate the dot product with a: (6i + 2j - 4k) ⋅ (i + j + 3k) = (6)(1) + (2)(1) + (-4)(3) = 6 + 2 - 12 = -4.
c) Cross product (vector product): a × (b + c) = (a × b) + (a × c).
Substituting the values, we get: a × (b + c) = (6i + 2j - 4k) × ((-2i - j + k) + (3i + 2j + 2k)).
Step 1: Add b and c: (-2i - j + k) + (3i + 2j + 2k) = i + j + 3k.
Step 2: Calculate the cross product with a: (6i + 2j - 4k) × (i + j + 3k) = (-4 - 2)i + (12 - 6)j + (2 - (-2))k = -6i + 6j + 4k.