The formula for vector addition is A + B = (A_x + B_x)i + (A_y + B_y)j + (A_z + B_z)k.
The magnitude of a vector is found by taking the square root of the sum of the squares of its components. In this case, |A + B| = sqrt[(3+(-1))^2 + (2+2)^2 + (1+B)^2] = 6. Expanding this equation, we get (2)^2 + (4)^2 + (1+B)^2 = 36. Solving for B, we get B = 3 or -5.
A vector in its component form is written as A = A_xi + A_yj + A_zk. This represents the vector's magnitude and direction in the x, y, and z directions respectively.
The i, j, and k unit vectors represent the standard unit vectors in the x, y, and z directions respectively. They are used to indicate the direction of a vector's components and are crucial in vector calculations and representations.
The dot product of two vectors A and B is calculated by multiplying their corresponding components and taking the sum of these products. In this case, A * B = (3*(-1)) + (2*2) + (1*B) = -3 + 4 + B = B + 1.