Answer:
λ = -37.9
Explanation:
The position vector of A is given by 41-2j, which means that point A has coordinates (41, -2) relative to the origin O. Similarly, the position vector of B is given by λi +2j, which means that point B has coordinates (λ, 2) relative to the origin O.
The vector AB is given by the difference between the position vectors of B and A:
AB = (λi + 2j) - (41 - 2j)
= (λi + 4j) - 41
The unit vector in the direction of AB is given as 0.31 + 0.4j. Since this is a unit vector, its length is equal to 1, which means we can find the length of AB as follows:
|AB| = |0.31 + 0.4j| * |(λi + 4j) - 41|
= 1 * |(λi + 4j) - 41|
= |(λi + 4j) - 41|
We know that the direction of AB is in the same direction as the vector (λi + 4j) - 41, which means that we can find a scalar multiple k such that:
0.31 + 0.4j = k * [(λi + 4j) - 41]
Expanding this equation, we get:
0.31 + 0.4j = kλi + 4kj - 41k
Since the left-hand side is a complex number, we must equate the real and imaginary parts separately. Equating the real parts gives:
0.31 = kλ - 41k
Equating the imaginary parts gives:
0.4 = 4kj
Since we want to find the value of λ, we can solve for k in the second equation:
k = 0.1/j = -0.1i
Substituting this value of k into the first equation, we get:
0.31 = -0.1λi + 4(-0.1i)(2) - 41(-0.1i)
Simplifying this equation, we get:
0.31 = -0.1λi - 0.8i + 4.1i
Equating the real and imaginary parts separately, we get:
-0.1λ = 3.79 (real parts)
-0.8 + 4.1 = 3.3i (imaginary parts)
Solving for λ, we get:
λ = -37.9
Therefore, the value of λ is -37.9.