Let u, v and w be vectors in R3. Which of the following calculations is possible? (a) u + (v · w) (b) u · (v × w) (c) (u · v) × w (d) u × (v · w)
Question
Let u, v and w be vectors in R3. Which of the following calculations is possible? (a) u + (v · w) (b) u · (v × w) (c) (u · v) × w (d) u × (v · w)
Solution
(a) u + (v · w): This calculation is not possible. The dot product of v and w (v · w) results in a scalar, and you cannot add a scalar to a vector (u).
(b) u · (v × w): This calculation is possible. The cross product of v and w (v × w) results in a vector, and the dot product of a vector (u) and this result is a scalar.
(c) (u · v) × w: This calculation is not possible. The dot product of u and v (u · v) results in a scalar, and you cannot take the cross product of a scalar and a vector (w).
(d) u × (v · w): This calculation is not possible. The dot product of v and w (v · w) results in a scalar, and you cannot take the cross product of a vector (u) and a scalar.
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