Explanation
1. Calculate the Magnitude of Vector A
Given A=2i^+j^−2k^:
∣A∣=22+12+(−2)2=4+1+4=9=3
2. Use the Magnitude of the Difference Vector
Given ∣C−A∣=22. Squaring both sides:
Substitute ∣A∣=3 and the given condition A⋅C=∣C∣:
3. Calculate the Cross Product A×B
Using the determinant method:
A×B=i^21amp;j^amp;1amp;1amp;k^amp;−2amp;0
A×B=i^(0−(−2))−j^(0−(−2))+k^(2−1)
4. Find the Magnitude of A×B
∣A×B∣=22+(−2)2+12=4+4+1=3
5. Calculate ∣(A×B)×C∣
The magnitude of the cross product between two vectors u and v is ∣u∣∣v∣sinϕ, where ϕ is the angle between them.
Let u=A×B and v=C. The angle ϕ=30∘.
∣(A×B)×C∣=∣A×B∣⋅∣C∣⋅sin30∘
Correct Option: (b) 3/2
