- $ (2B) \times (3C) $ – $ B \times C $
- $ \overrightarrow{A} \times ( \overrightarrow{B} \times \overrightarrow{C} ) $
- If v1 and v2 are perpendicular, | v1, v2 |
- 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).
\[ A \times B = (1, 0, 1) \times (0, 1, 0) \]
\[ A \times B = \begin {vmatrix} i & j & k \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end {vmatrix} \]
\[ A \times B = (-1, 0, 1) \]