The expression (I + J) represents the vector sum of the standard unit vectors in the Cartesian coordinate system, where I represents the unit vector in the x-direction and J represents the unit vector in the y-direction. To determine the magnitude and direction of (I + J), we can consider it as the result of adding the components of I and J.
The unit vector I has a magnitude of 1 in the x-direction (horizontal) and a direction of 0 degrees (or along the positive x-axis). The unit vector J has a magnitude of 1 in the y-direction (vertical) and a direction of 90 degrees (or along the positive y-axis).
When we add I and J, we are essentially adding their components. The x-component of (I + J) is 1 (from I), and the y-component is 1 (from J). Thus, the vector (I + J) has components (1, 1).
To find the magnitude of (I + J), we use the Pythagorean theorem, which states that the magnitude (or length) of a vector in two dimensions is given by the square root of the sum of the squares of its components. In this case:
Magnitude = sqrt((1^2) + (1^2)) = sqrt(2)
So, the magnitude of (I + J) is sqrt(2).
To determine the direction of (I + J), we can use trigonometry. The direction (or angle) of a vector is usually measured counterclockwise from the positive x-axis. In this case, we can calculate the angle using the inverse tangent (arctan) function:
Direction = arctan(y-component / x-component) = arctan(1/1) = arctan(1) ≈ 45 degrees
Therefore, the magnitude of (I + J) is sqrt(2), and its direction is approximately 45 degrees counterclockwise from the positive x-axis.