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Let vectors A =(2, -1, -4), B =(−1, 0, 2), and C =(3, 4, 1). Calculate the following expressions for these vectors:

  1. $ (2B) \times (3C) $ – $ B \times C $
  2. $ \overrightarrow{A} \times ( \overrightarrow{B} \times \overrightarrow{C} ) $
  3. If v1 and v2 are perpendicular, | v1, v2 |
  4. If v1 and v2 are parallel, | v1 , v2 |

This question aims to find the cross-product of three different vectors in different scenarios.

This question is based on the concept of vector multiplication, especially the cross-product of vectors. Cross-product of vectors is the multiplication of vectors, resulting in a third vector perpendicular to both vectors. It is also called a vector product. If we have A and B as two vectors, then:

\[ A \times B = \begin {vmatrix} i & j & k \\ a1 & a2 & a3 \\ b1 & b2 & b3 \end {vmatrix} \]

Expert Answer

We can calculate these vectors by taking their cross-products.

a) $ (2B) \times (3C) $

\[ 2B = 2 \times (-1, 0, 2) \]

\[ 2B = (-2, 0, 4) \]

\[ 3C = 3 \times (3, 4, 1) \]

\[ 3C = (9, 12, 3) \]

\[ (2B) \times (3C) = (-2, 0, 4) \times (9, 12, 3) \]

\[ 2B) \times (3C) = \begin {vmatrix} i & j & k \\ -2 & 0 & 4 \\ 9 & 12 & 3 \end {vmatrix} \]

Simplifying the determinant of the matrix, we get:

\[ (2B) \times (3C) = (-48, 42, -24) \]

b) $ B \times C $

\[B \times C = ( -1, 0, 2 ) \times ( 3, 4, 1 ) \]

\[ B \times C = \begin {vmatrix} i & j & k \\ -1 & 0 & 2 \\ 3 & 4 & 1 \end {vmatrix} \]

Simplifying the determinant of the matrix, we get:

\[ B \times C = ( -8, 7, 4 ) \]

c) $ \overrightarrow{A} \times ( \overrightarrow{B} \times \overrightarrow{C} ) $

We already calculated B x C in the previous part. Now we take the cross-product of A with the result of B x C.

\[ A \times ( B \times C ) = ( 2, -1, -4 ) \times ( -8, 7, 4 ) \]

\[ A \times ( B \times C ) = \begin {vmatrix} i & j & k \\ 2 & -1 & -4 \\ -8 & 7 & 4 \end {vmatrix} \]

Simplifying the determinant of the matrix, we get:

\[ A \times ( B \times C ) = ( 24, 24, 6 ) \]

d) If we have two perpendicular vectors $v_1$ and $v_2$ and we need to find their cross-product, we can use the following formula.

\[ v1 \times v2 = v1 v2 \sin \theta \]

\[ v1 \times v2 = v1 v2 \sin ( 90^ {\circ} ) \]

\[ v1 \times v2 = v1 v2 (1) \]

\[ v1 \times v2 = v1 v2 \]

e) If we have two parallel vectors $v_1$ and $v_2$ and need to find their cross-product, we can use the following formula.

\[ v1 \times v2 = v1 v2 \sin \theta \]

\[ v1 \times v2 = v1 v2 \sin ( 0^ {\circ} ) \]

\[ v1 \times v2 = v1 v2 (0) \]

\[ v1 \times v2 = 0 \]

Numerical Result

a) $ (2B) \times (3C) = (-48, 42, -24) $

b) $ B \times C = ( -8, 7, 4 ) $

c) $ A \times ( B \times C ) = ( 24, 24, 6 ) $

d) $ v1 \times v2 = v1 v2 $

e) $ v1 \times v2 = 0 $

Example

Find the cross-product of vectors A (1, 0, 1) and B(0, 1, 0).

\[ A \times B = (1, 0, 1) \times (0, 1, 0) \]

\[ A \times B = \begin {vmatrix} i & j & k \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end {vmatrix} \]

\[ A \times B = (-1, 0, 1) \]

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