Question

The angle of vector sum of two vectors a and b is 8unit and the magnitude of b is 5unit if the direction of a is reversed the magnitude of the sum of vector 10 unit find the angle that a and b makes with each other

Answer 1

Answer: The angle between vectors a and b is approximately 65.5 degrees

Explanation: Let’s first calculate the angle between vectors a and b. We know that the angle between the vector sum of two vectors a and b is 8 units. Using the cosine rule, we can calculate the magnitude of the vector sum of a and b:

∣a+b∣2=∣a∣2+∣b∣2+2∣a∣∣b∣cosθ

where θ is the angle between vectors a and b. We know that ∣b∣=5 units. If we reverse the direction of vector a, the magnitude of the vector sum becomes 10 units. Using the same formula, we can solve for θ:

102100∣a∣2−10∣a∣cosθ+75(∣a∣−5cosθ)2∣a∣​=∣a∣2+52−2∣a∣5cosθ=∣a∣2+25−10∣a∣cosθ=0=0=5cosθ​

Substituting this into the previous equation:

826439cosθ​=(5cosθ)2+52−2(5cosθ)(5)cosθ=25cos2θ−50cos2θ=25cos2θ=±539​​​

Since the angle between vectors a and b is acute, we take the positive value of cosθ:

θ=cos−1(539​​)≈1.14 rad or 65.5∘​

Therefore, the angle between vectors a and b is approximately 65.5 degrees.