Mathematics: Find v × w if v = 3i + 8j – 6k and w = –4i – 2j – 3k. a. –2i + 8j – 3k b. –36i + 33j + 26k c. 33i + 26j + 8k d. 3i – 2j + 8k

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Find v × w if v = 3i + 8j – 6k and w = –4i –

Ответ:

kitttimothy55

05.01.2022

Using the determinant method, the cross product is

$\begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\3&8&-6\\-4&-2&-3\end{vmatrix}=\begin{vmatrix}8&-6\\-2&-3end{vmatrix}\,\vec\imath-\begin{vmatrix}3&-6\\-4&-3\end{vmatrix}\,\vec\jmath+\begin{vmatrix}3&8\\-4&-2\end{vmatrix}\,\vec k=-36\,\vec\imath+33\,\vec\jmath+26\,\vec k$

so the answer is B.

Or you can apply the properties of the cross product. By distributivity, we have

(3i + 8j - 6k) x (-4i - 2j - 3k)

= -12(i x i) - 32(j x i) + 24(k x i) - 6(i x j) - 16(j x j) + 12(k x j) - 9(i x k) - 24(j x k) + 18(k x k)

Now recall that

(i x i) = (j x j) = (k x k) = 0 (the zero vector)(i x j) = k(j x k) = i(k x i) = j(a x b) = -(b x a) for any two vectors a and b

Putting these rules together, we get

(3i + 8j - 6k) x (-4i - 2j - 3k)

= -32(-k) + 24j - 6k + 12(-i) - 9(-j) - 24i

= (-12 - 24)i + (24 + 9)j + (32 - 6)k

= -36i + 33j + 26k

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Find v × w if v = 3i + 8j – 6k and w = –4i – 2j – 3k.

a. –2i + 8j – 3k
b. –36i + 33j + 26k
c. 33i + 26j + 8k
d. 3i – 2j + 8k

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