Vectors (11Acd05)

Sheet 02

1. What will be the horizontal and vertical component of a \(50 N\) force that is \(60^{o}\) above the horizontal?
   
   Ans: \(F_{X}=25N\) and \(F_{Y}=25\sqrt{3}N\).
   
   
2. Find, the unit vector parallel to the resultant of vectors \(\vec{A}=\hat{i}+4\hat{j}-\hat{k}\) and \(\vec{B}=3\hat{i}-5\hat{j}+\hat{k}\).
   
   Ans: \(\hat{R}=\frac{1}{3\sqrt{2}}\left(4\hat{i}-\hat{j}-\hat{k}\right)\).
   
   
3. Find, the value of \(\lambda\) in the unit vector, given as \(0.4\hat{i}+0.8\hat{j}+\lambda\hat{k}\).
   
   Ans: \(\lambda=\sqrt{0.2}\).
   
   
4. Three co-planner vectors are given by, \(\vec{A}=4\hat{i}-\hat{j}\), \(\vec{B}=-3\hat{i}+2\hat{j}\) and \(\vec{C}=-3\hat{j}\). Find, the magnitude of sum of three vectors.
   
   Ans: \(|\vec{R}|=\sqrt{5}\).
   
   
5. Two billiard balls are rolling on a flat table. One has velocity component \(V_{x}= 1 ms^{-1}\), \(V_{y}=\sqrt{3} ms^{-1}\) and other has components \(\text{V}^{‘}_{x}= 2 ms^{-1}\) and \(\text{V}^{‘}_{y}= 2 ms^{-1}\). If, both balls strat moving from same point, what is the angel between their paths?
   
   Ans: \(15^{o}\).
   
   
6. A child pulls a rope attached to a stone with a force of \(60 N\). The rope makes an angel of \(30^{o}\) with the ground. Then Calculate: (a) Effective value of pull tending to move the stone along the ground? (b) The force tending to lift the stone vertically upwards?
   
   Ans: \(F_{X}=30\sqrt{3}N\) and \(F_{Y}=30N\).
   
   
7. A car of mass \(1000 Kg\) is resting on an inclined plane making an angel \(30^{o}\) with horizontal. What is the weight of the car? Also, Find horizontal and vertical component of the weight of the car?
   
   Ans: \(10kN\), \(8.66kN\) and \(5kN\).
   
   
8. If, \(\vec{A}= 3\hat{i}+4\hat{j}\) and \(\vec{B}=\hat{i}+24\hat{j}\). Find, the vector having the same magnitude as \(\vec{B}\) and parallel to \(\vec{A}\).
   
   Ans: \(\vec{C}=15\hat{i}+20\hat{j}\).
   
   
9. Find, the unit vector parallel to the resultant of vectors \(\vec{A}=2\hat{i}-6\hat{j}-3\hat{k}\) and \(\vec{B}=4\hat{i}+3\hat{j}-\hat{k}\).
   
   Ans: \(\frac{1}{\sqrt{61}}\left(6\hat{i}-3\hat{j}-4\hat{k}\right)\).
   
   
10. If, \(\vec{A}=2\hat{i}+4\hat{j}-5\hat{k}\), then find magnitude of \(\vec{A}\) and direction cosine of vector \(\vec{A}\).
    
    Ans: \(\sqrt{45}\), \(\frac{2}{\sqrt{45}}\), \(\frac{4}{\sqrt{45}}\) and \(\frac{-5}{\sqrt{45}}\).
    
    
11. If, the vectors \(\vec{A}=2\hat{i}+5\hat{j}-\hat{k}\), \(\vec{B}=5\hat{i}+Y\hat{j}+\hat{k}\) and \(\vec{C}=\hat{i}+2\hat{j}+2\hat{k}\) are co-planer. Then the value of Y?
    
    Ans: \(Y=7\).
    
    
12. Three forces \(\vec{A}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{B}=2\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{c}\) are acting on a body which is kept in equlibrium. Find, \(\vec{C}\)?
    
    Ans: \(\vec{C}=-\left(3\hat{i}+4\hat{k}\right)\)
    
    
13. At what angel two force vectors of magnitude \(2F\) and \(\sqrt{2}F\) acts so that the resultant of two forces is \(\sqrt{10}F\)
    
    Ans: \(45^{o}\).
    
    
14. Calculate: the area of the triangle determined by the two vectors \(\vec{A}=3\hat{i}+4\hat{j}\) and \(\vec{B}=-3\hat{i}+7\hat{j}\).
    
    Ans: \(\frac{33}{2}Sq.Units\)
    
    
15. Calculate: the area of the parallelogram when adjacent sides are given by the vectors \(\vec{A}=\hat{i}+2\hat{j}+3\hat{k}\) and \(\vec{B}=2\hat{i}-3\hat{j}+\hat{k}\).
    
    Ans: \(\sqrt{195}Sq.Unit\).
    
    
16. What is the angle between \(\hat{i}+\hat{j}+\hat{k}\) and \(\hat{i}\).
    
    Ans: \(\theta=Cos^{-1}\frac{1}{\sqrt{3}}\).