 # 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:

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

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).

$A \times B = (1, 0, 1) \times (0, 1, 0)$

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

$A \times B = (-1, 0, 1)$