To find the angle between two vectors, you can use the dot product formula and the magnitude of the vectors. Let's calculate it:
Vector A = I + j + l Vector B = b - 2I - 2j - 2k
The dot product of two vectors A and B is given by:
A · B = |A| |B| cosθ
where |A| and |B| are the magnitudes of vectors A and B, respectively, and θ is the angle between them.
First, let's find the magnitudes of the vectors A and B:
|A| = √(1^2 + 1^2 + l^2) = √(2 + l^2) |B| = √(b^2 + (-2)^2 + (-2)^2 + (-2)^2) = √(b^2 + 12)
Next, let's calculate the dot product of A and B:
A · B = (1)(-2) + (1)(-2) + (l)(-2) = -2 - 2l
Now, we can substitute the values into the dot product formula:
-2 - 2l = (√(2 + l^2))(√(b^2 + 12)) cosθ
To find the angle θ, we can rearrange the equation:
cosθ = (-2 - 2l) / (√(2 + l^2))(√(b^2 + 12))
The angle θ can be determined by taking the arccosine (inverse cosine) of both sides:
θ = arccos[(-2 - 2l) / (√(2 + l^2))(√(b^2 + 12))]
Note that the specific value of the angle θ will depend on the values of l and b that you substitute into the equation.