Question

Calculate the cross product assuming that u × w = ⟨4, −5, 6⟩. (−3u − w) × w.

a) ⟨0, 0, 0⟩
b) ⟨12, −15, 18⟩
c) ⟨−12, 15, −18⟩
d) ⟨−4, 5, −6⟩

Answer 1

The cross product of (-3u - w) × w, given that u × w equals ⟨4, -5, 6⟩, is ⟨-12, 15, -18⟩, making option c) the correct answer. Therefore, the cross product of (-3u - w) × w is ⟨-12, 15, -18⟩, which corresponds to option c).

Step-by-step explanation:

To calculate the cross product of (-3u - w) × w, we can use the distributive property of cross products and the anticommutative property. Since we already know that u × w = ⟨4, -5, 6⟩, we can calculate:

(-3u - w) × w = -3(u × w) - (w × w)

The cross product of a vector with itself is always ⟨0, 0, 0⟩, thus w × w = ⟨0, 0, 0⟩. Then the equation simplifies to:

(-3u - w) × w = -3(u × w)

Substitute the known u × w:

-3(⟨4, -5, 6⟩) = ⟨-12, 15, -18⟩

To calculate the cross product of (−3u − w) × w, we can use the formula Č = Ả × B = (Ay B₂ – Az By)Î + (Az Bx − Ax Bz)Ĵ + (Ax By –− AyBx)Ê.

Given that u × w = ⟨4, −5, 6⟩, we can substitute the values into the formula to find the cross product:

Č = (−3)(4)−(−5)(0)Î + (−5)(0)−(−3)(6)Ĵ + (−3)(0)−(−5)(6)Ê

After simplifying the equation, we get Č = ⟨12, −15, 18⟩.

Therefore, the cross product of (-3u - w) × w is ⟨-12, 15, -18⟩, which corresponds to option c).