For the following three vectors, what is 3C·(2AxB)?
A = 2.00ihat + 3.00jhat - 5.00khat
B= -3.00ihat + 5.00jhat + 2.00khat
C = 9.00ihat - 8.00jhat
(924,34,0)
To find the value of 3C · (2A × B), we need to perform vector operations - cross product and dot product. Here's how you can do it step by step:
1. Find the cross product of A and B: A × B
- To calculate the cross product, use the determinant method or the cross product formula:
A × B = (AyBz - AzBy)ihat + (AzBx - AxBz)jhat + (AxBx - AyBx)khat
- Substitute the values from the given vectors:
A × B = ((3)(2) - (-5)(5))ihat + ((-5)(-3) - (2)(2))jhat + ((2)(-3) - (3)(-5))khat
2. Simplify the cross product result:
A × B = (6 + 25)ihat + (15 - 4)jhat + (-6 + 15)khat
A × B = 31ihat + 11jhat + 9khat
3. Multiply the cross product by 2A:
2A × B = 2(31ihat + 11jhat + 9khat)
= 62ihat + 22jhat + 18khat
4. Multiply the result by 3C:
3C · (2A × B) = 3(9ihat - 8jhat) · (62ihat + 22jhat + 18khat)
= (27ihat - 24jhat) · (62ihat + 22jhat + 18khat)
5. Calculate the dot product:
Recall that the dot product is calculated by multiplying the corresponding components and adding them together.
3C · (2A × B) = (27 * 62) + (-24 * 22) + (0 * 18)
= 1674 - 528
= 1146
Therefore, 3C · (2A × B) is equal to 1146.