Vectors (11Acd05)

Sheet 04 (Vector Products)

1. Prove That: \(\left(\vec{A}+2\vec{B}\right).\left(2\vec{A}-3\vec{B}\right)=2A^{2}+AB\cos\theta-6B^{2}\).
   
   
2. If, \(|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|\) determine the angle between \(\vec{a}\) and \(\vec{b}\), making use of the properties of dot product of vectors.
   
   Ans: \(\theta=90^{o}\).
   
   
3. If, three vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) have magnitudes \(8, 15\) and \(17\) units respectively and \(\vec{A}+\vec{B}=\vec{C}\). Find: the angle between \(\vec{A}\) and \(\vec{B}\).
   
   Ans: \(90^{o}\).
   
   
4. Show that: \(\left(\vec{A}-\vec{B}\right)\times\left(\vec{A}+\vec{B}\right)=2\left(\vec{A}\times\vec{B}\right)\).
   
   
5. The diagonals of a parallelogram are given by \(\vec{R_{1}}=3\hat{i}+2\hat{j}-7\hat{k}\) and \(\vec{R_{2}}=5\hat{i}+6\hat{j}-3\hat{k}\).Find the area of parallelogram.
   
   Ans: \(\sqrt{509}Units\).
   
   
6. Find; \(|\vec{A}\times\vec{B}|\). If, \(|\vec{A}|=10\), \(|\vec{B}|=2\) and \(\vec{A}.\vec{B}=12\).
   
   Ans: \(|\vec{A}\times \vec{B}|=16\).
   
   
7. If, \(\vec{A}=\vec{B}-\vec{C}\) then determine the angle between \(\vec{A}\) and \(\vec{B}\).
   
   Ans: \(\theta=Cos^{-1}\frac{A^{2}+B^{2}-C^{2}}{}\)
   
   
8. If, vector \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) have magnitudes \(5,12\) and \(13\) units respectively and \(\vec{A}+\vec{B}=\vec{C}\).Find the angle between \(\vec{B}\) and \(\vec{C}\).
   
   Ans: \(\theta=Cos^{-1}\left(\frac{12}{13}\right)\).
   
   
9. If, \(\vec{A}+\vec{B}=\vec{C}\) and \(A^{2}+B^{2}=C^{2}\). Then, prove that \(\vec{A}\) and \(\vec{B}\) are perpendicular to each other.
   
   
10. For, any two vectors \(\vec{A}\) and \(\vec{B}\) prove that; \(\left(|\vec{A}\times\vec{B}|\right)^{2}=A^{2}B^{2}-\left(\vec{A}.\vec{B}\right)^{2}\).
    
    
11. For any three vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\). Prove that: \(\vec{A}\times\left(\vec{B}+\vec{C}\right)+\vec{B}\times\left(\vec{C}+\vec{A}\right)+\vec{C}\times\left(\vec{A}+\vec{B}\right)=0\).
    
    
12. If, \(\vec{A}=-2\hat{i}+\hat{j}-3\hat{k}\) and \(\vec{B}=5\hat{i}+2\hat{j}-\hat{k}\). Then, find \(\left(\vec{A}+\vec{B}\right).\left(\vec{A}-\vec{B}\right)\).
    
    Ans: \(-16\).
    
    
13. If, \(\vec{A}\) and \(\vec{B}\) are two vectors and \(\theta\) be the angle between them. Find, the value of \(\left(\vec{B}\times\vec{A}\right).\vec{A}\)
    
    Ans: \(Zero\left(0\right)\).
    
    
14. If, \(\vec{A}\) and \(\vec{B}\) are two vectors and \(\theta\) be the angle between them. If \(|\vec{A}\times\vec{B}|=\sqrt{3}\left(\vec{A}.\vec{B}\right)\). Then what is the value of \(\theta\).
    
    Ans: \(\frac{\pi}{3}\).
    
    
15. Given; \(\vec{A}+\vec{B}=\vec{C}\) and \(C=\left[\vec{C}.\vec{C}\right]^{\frac{1}{2}}\). Show that \(C=\left(A^{2}+B^{2}+2AB\cos\theta\right)^{\frac{1}{2}}\) where \(\theta\) be the angle between \(\vec{A}\) and \(\vec{B}\).
    
    
16. Prove that \(|\vec{A}+\vec{B}|=\sqrt{A^{2}+B^{2}}\). If, \(\vec{A}\) and \(\vec{B}\) are perpendicular to each other.