Vectors (11Acd05)

Sheet 03 (Vector Products)

1. Prove, that the vectors \(\vec{A}=\hat{i}+2\hat{j}+3\hat{k}\) and \(\vec{B}=2\hat{i}-\hat{j}\) are perpendicular to each other.
   
   
2. For what value of \(m\), is the \(\vec{A}=2\hat{i}+3\hat{j}-6\hat{k}\) perpendicular to the vector \(\vec{B}=3\hat{i}-m\hat{j}+6\hat{k}\)?
   
   Ans: \(m=-10\).
   
   
3. If, a force of \(\vec{F}=\left(5\hat{i}+4\hat{j}\right)\) \(N\) displaces a body through \(\vec{S}=\left(3\hat{i}+4\hat{j}\right)\) \(m\) in \(5 Sec\). Then, find Power?
   
   Ans: \(Power=6.17Watt\).
   
   
4. Under a force of \(\left(10\hat{i}-3\hat{j}+6\hat{k}\right)\)\(N\), a body of mass \(5 Kg\) is displaced from the position \(\left(6\hat{i}+5\hat{j}-3\hat{k}\right)\) to the position \(\left(10\hat{i}-2\hat{j}+7\hat{k}\right)\). Calculate the work done.
   
   Ans: \(Work=121J\).
   
   
5. Find, the angel between the vectors, \(\vec{A}=\hat{i}+2\hat{j}-\hat{k}\) and \(\vec{B}=-\hat{i}+\hat{j}-2\hat{k}\).
   
   Ans: \(\theta=60^{o}\).
   
   
6. If, the magnitude of the two vectors are 2 and 3 and the magnitude of their scalar product is \(3\sqrt{2}\), then what should be the angel between them?
   
   Ans: \(\theta=45^{o}\).
   
   
7. Find, the angel between force \(\vec{F}=\left(3\hat{i}+4\hat{j}-5\hat{k}\right)\) Unit and Displacement \(\vec{S}=\left(5\hat{i}+4\hat{j}+3\hat{k}\right)\) Unit. Also, find the projection of \(\vec{F}\) on \(\vec{S}\).
   
   Ans: \(\theta=Cos^{-1}\left(0.32\right)=71.34^{o}\) and \(0.32\left(5\hat{i}+4\hat{j]+3\hat{k}\right)\).
   
   
8. Determine the angles which the vector \(\vec{A}=\left(5\hat{i}+0\hat{j}+5\hat{k}\right)\) makes with X,Y and Z axes.
   
   Ans: \(\alpha=45^{o}\), \(\beta=90^{o}\) and \(\gamma=45^{o}\).
   
   
9. Find, the component of vector, \(\vec{A}=3\hat{i}+4\hat{j}\) along the direction of vector, \(\vec{B}=2\hat{i}-3\hat{j}\).
   
   Ans: \(-\frac{6}{13}\left(2\hat{i}-3\hat{j}\right)\).
   
   
10. Find, the unit vector perpendicular to the vectors, \(\vec{A}=4\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{B}=-2\hat{i}+\hat{j}-2\hat{k}\).
    
    Ans: \(\hat{C}=\frac{1}{3}\left(-\hat{i}+2\hat{j}+2\hat{k}\right)\).
    
    
11. Determine, the sine of angle between vectors \(\vec{A}=3\hat{i}-4\hat{j}+5\hat{k}\) and \(\vec{B}=\hat{i}-\hat{j}+\hat{k}\).
    
    Ans: \(Sin\theta=\farc{1}{5}\).
    
    
12. Find, the vector whose magnitude is \(7\) and which is perpendicular to each vectors. \(\vec{A}=2\hat{i}-3\hat{j}+6\hat{k}\) and \(\vec{B}=\hat{i}+\hat{j}-\hat{k}\).
    
    Ans: \(\frac{1}{\sqrt{2}}\left(-3\hat{i}+8\hat{j}+5\hat{k}\right)\).
    
    
13. If, \(\vec{A}\) and \(\vec{B}\) are two such vectors that \(|\vec{A}|=2\), \(|\vec{B}|=7\) and \(\vec{A}\times\vec{B}=3\hat{i}+2\hat{j}+6\hat{k}\), Find, the angle between \(\vec{A}\) and \(\vec{B}\).
    
    Ans: \(\theat=\frac{\pi}{6}\).
    
    
14. Find, the moment about the point \(\left(\hat{i}+2\hat{j}-\hat{k}\right)\) of a force represented by \(3\hat{i}+\hat{k}\) acting through the point \(\left(2\hat{i}-\hat{j}+3\hat{k}\right)\).
    
    Ans: \(\vec{\tau}=\left(-3\hat{i}+11\hat{j}+9\hat{k}\right)\).
    
    
15. Two forces each of magnitude \(F/2\), acts at right angle. Their effect may be neutralized by a third force acting bisector in the opposite direction. What is the magnitude of the third force?
    
    Ans: \(\frac{F}{\sqrt{2}}\)
    
    
16. The sum of magnitudes of two vectors is \(18\). The magnitude of their resultant is \(12\). If, the resultant is perpendicular to the one of vectors, then find the magnitudes of the two vectors.
    
    Ans: \(P=5\) and \(Q=13\).
    
    
17. The linear velocity of a rotating body is given by \(\vec{v}=\vec{\omega}\times\vec{r}\). If, \(\vec{\omega}=\hat{i}-2\hat{j}+\hat{k}\) and \(\vec{r}=4\hat{j}-3\hat{k}\). Then, What is the magnitude of \(\vec{v}\).
    
    Ans: \(\sqrt{29}Units\).