Explain with the examples a examples of a cross product of a vectors
The cross product is an operation between two vectors that results in a vector perpendicular to both of the original vectors. It is denoted by the symbol "×" or "⨯".
Let's consider two vectors, A and B, in three-dimensional space:
1. Vector A: A = (2, 3, 1)
Vector B: B = (4, -1, 5)
To find the cross product of A and B, we use the determinant formula:
A × B = | i j k |
| 2 3 1 |
| 4 -1 5 |
Calculating the determinants of the 2x2 matrices:
A × B = (3 * 5 - 1 * -1) i - (2 * 5 - 1 * 4) j + (2 * -1 - 3 * 4) k
= 16i + 6j - 14k
Therefore, A × B = (16, 6, -14)
The resulting vector (16, 6, -14) is perpendicular to both A and B.
2. Vector A: A = (1, 2, 3)
Vector B: B = (4, 5, 6)
A × B = | i j k |
| 1 2 3 |
| 4 5 6 |
A × B = (2 * 6 - 3 * 5) i - (1 * 6 - 3 * 4) j + (1 * 5 - 2 * 4) k
= -3i -2j + 3k
Therefore, A × B = (-3, -2, 3)
The resulting vector (-3, -2, 3) is perpendicular to both A and B.
In summary, the cross product of two vectors results in a new vector that is perpendicular to both of the original vectors.
State the rules when finding a cross product of vectors
When finding the cross product of two vectors, there are certain rules to follow:
1. The cross product is only defined for vectors in three-dimensional space. It cannot be computed for vectors in two-dimensional space.
2. The cross product of two vectors, A × B, is a vector perpendicular to both A and B.
3. The magnitude (or length) of the cross product vector is given by the product of the magnitudes of A and B multiplied by the sine of the angle θ between them: |A × B| = |A| |B| sin(θ).
4. The direction of the cross product is determined by the right-hand rule: if you curl the fingers of your right hand from vector A towards vector B, the direction your thumb points will be the direction of the cross product.
5. The cross product is anti-commutative, meaning that changing the order of the vectors will result in a vector with the opposite direction: A × B = - (B × A).
6. If the two vectors A and B are parallel or collinear (meaning they lie on the same line or in the same direction), then their cross product will be zero: A × B = 0.
7. The cross product is distributive over addition: (A + B) × C = (A × C) + (B × C).
8. The cross product is not associative: A × (B × C) is not equal to (A × B) × C. The results will differ by a scalar factor and direction.
These rules govern the computation and properties of the cross product of vectors.
Momentum is the vector is the vector quantity and can be fund by multiplying the mass and velocity (P=M×V) a mass of (2i -j+ 3k) ky and velocity of (7i+2j-4k) m/s . Find the cross product of mass and velocity
To find the cross product of the mass and velocity vectors, we can use the determinant formula:
P = M × V = | i j k |
| 2 -1 3 |
| 7 2 -4 |
Calculating the determinant of the 2x2 matrices:
P = ( -1 * -4 - 3 * 2 ) i - (2 * -4 - 3 * 7) j + (2 * 2 - (-1) * 7) k
= (4 - 6) i - (-8 - 21) j + (4 + 7) k
= -2i - (-29)j + 11k
= -2i + 29j + 11k
Therefore, the cross product of the mass and velocity vectors is -2i + 29j + 11k.
A: = 3i+20k, b:= 4j, c= 2i +3j+ 7k. Find : (1) A×B (2). Axc (3) ( A×c× )b
To find the cross product of vectors A and B, we use the determinant formula:
1) A × B = | i j k |
| 3 0 20 |
| 0 4 0 |
Calculating the determinants of the 2x2 matrices:
A × B = (0 * 0 - 20 * 4) i - (3 * 0 - 20 * 0) j + (3 * 4 - 0 * 0) k
= -80i + 0j + 12k
= -80i + 12k
Therefore, A × B = -80i + 12k
To find the cross product of vectors A and C:
2) A × C = | i j k |
| 3 0 20 |
| 2 3 7 |
Calculating the determinants of the 2x2 matrices:
A × C = (0 * 7 - 20 * 3) i - (3 * 7 - 2 * 20) j + (3 * 3 - 0 * 2) k
= -60i - 41j + 9k
Therefore, A × C = -60i - 41j + 9k
To find the cross product of (A × C) and B:
3) (A × C) × B = | i j k |
| -60 -41 9 |
| 0 4 0 |
Calculating the determinants of the 2x2 matrices:
(A × C) × B = (-41 * 0 - 9 * 4) i - (-60 * 0 - 9 * 0) j + (-60 * 4 - (-41) * 0) k
= 0i - 0j - 240k
= -240k
Therefore, (A × C) × B = -240k.