You should write a clear and concise explanation for your assignments. You should always provide justification or working for any results that you obtain unless these results are trivial. When you apply a theorem, state the theorem and make sure you check and state if all conditions of the theorem are satisfied before using it. You should avoid giving unnecessary detail or irrelevant information. Choose the best solution for the following question. Calculate where \begin{equation*}\mathbf{u}=\begin{pmatrix}2\0\1\end{pmatrix}\, \mbox{and}\, \mathbf{v}=\begin{pmatrix}-3\1\0\end{pmatrix}.\end{equation*}

Question

BlackTom AI · Accepted Answer

The cross product in R^3 is defined for vectors u = (a1,a2,a3) and v = (b1,b2,b3) by
u × v = (a2 b3 − a3 b2, a3 b1 − a1 b3, a1 b2 − a2 b1).
This formula is valid because u and v are both 3-dimensional vectors (so all conditions for the cross product are satisfied).

Step-by-step computation for u = (2,0,1), v = (−3,1,0):

1. Compute the first component: a2 b3 − a3 b2 = 0·0 − 1·1 = 0 − 1 = −1.
2. Compute the second component: a3 b1 − a1 b3 = 1·(−3) − 2·0 = −3 − 0 = −3.
3. Compute the third component: a1 b2 − a2 b1 = 2·1 − 0·(−3) = 2 − 0 = 2.

Thus u × v = (−1, −3, 2).

Verification (why this is correct): the cross product must be orthogonal to both u and v. Check dot products:
u · (u × v) = (2)(−1) + (0)(−3) + (1)(2) = −2 + 0 + 2 = 0.
v · (u × v) = (−3)(−1) + (1)(−3) + (0)(2) = 3 − 3 + 0 = 0.
Both dot products are zero, confirming orthogonality. The signs and components also match direct expansion of the 3×3 determinant |i j k; 2 0 1; −3 1 0|.

Brief note on other possible approaches (why they would be incorrect): using the dot product would give a scalar, not a vector, so it cannot replace the cross product. Computing v × u instead of u × v would produce the negation (1,3,−2), which is different due to anticommutativity of the cross product. Common sign errors arise from incorrect determinant expansion; the stepwise component formula above avoids that.

Conclusion: Using the formula for the cross product yields u × v = (−1, −3, 2), so option b (Using the formula for the cross product) is the correct choice.

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Let 𝐴 = ( 1 , 5 , − 2 ) , 𝐵 = ( 1 , 1 , 2 ) , 𝐶 = ( 3 , 2 , 5 ) be points in 𝑅 3 . Which of the following vectors is orthogonal to both 𝐴 𝐵 → and 𝐴 𝐶 → ?

A vector that is orthogonal (perpendicular) to both vectors 𝑎 _ = ⟨ − 2 , − 5 , 1 ⟩ and 𝑏 _ = ⟨ 1 , 3 , − 2 ⟩ is: Hint: Vectors 𝑣 _ and 𝑤 _ are orthogonal if and only if 𝑣 _ ⋅ 𝑤 _ = 0 .

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