Question

Let U=(K,2) And V(3,5). Find K Such That The Angle Between U And V Is π/3

Answer 1

The value of K for which the angle between vector U and V is π/3 can be found using the dot product and magnitude of vectors, and subsequently by solving a quadratic equation.

Step-by-step explanation:

To find the value of K such that the angle between vectors U=(K,2) and V=(3,5) is π/3 (or 60 degrees), we need to use the dot product of vectors which is defined as U · V = |U||V|cos(θ), where |U| and |V| are the magnitudes of vectors U and V, respectively, and θ is the angle between them.

Firstly, we calculate the dot product U · V:

• 
  
• (K,2) · (3,5) = K*3 + 2*5
• 
  
• U · V = 3K + 10

Next, we calculate the magnitudes of U and V:

• 
  
• |U| = √(K^2 + 2^2)
• 
  
• |V| = √(3^2 + 5^2) = √(9 + 25) = √34

Now we use the equation U · V = |U||V|cos(π/3), substituting the known values and solving for K:

1. 
   
2. 3K + 10 = (√(K^2 + 4))(√34)cos(π/3)
3. 
   
4. 3K + 10 = (√(K^2 + 4))(√34)(1/2)
5. 
   
6. 6K + 20 = (√(K^2 + 4))(√34)
7. 
   
8. 36K^2 + 240K + 400 = (K^2 + 4)(34)
9. 
   
10. 36K^2 + 240K + 400 = 34K^2 + 136
11. 
    
12. 2K^2 + 240K + 264 = 0
13. 
    
14. K^2 + 120K + 132 = 0
15. 
    
16. Solving this quadratic equation, we get two possible values for K

Therefore, the possible values of K can be found by factoring or using the quadratic formula.