Resultant Vector of 2i5j: Magnitude and Direction Explained

The vector can be understood as a combination of a movement of 2 units along the x-axis and -5 units along the y-axis. This vector is already in its resultant form, representing a single vector in a two-dimensional Cartesian coordinate system.

Magnitude of the Resultant Vector

The magnitude of a vector can be calculated using the formula:

(mathbf{v} sqrt{x^2 y^2})

For the vector , we substitute the values of the components:

(mathbf{v} sqrt{2^2 (-5)^2} sqrt{4 25} sqrt{29})

Therefore, the magnitude of the vector is (sqrt{29}). This value indicates the length of the vector in the coordinate system.

Direction of the Resultant Vector

The direction of the vector can be determined by calculating the angle (theta) with respect to the positive x-axis. This angle can be found using the arctangent function:

(theta arctanleft(frac{y}{x}right) arctanleft(frac{-5}{2}right))

Since y is negative, the vector is pointing towards the fourth quadrant, which means the angle will be in the range (pi

Adding Vectors 2hati and 5hatj

To find the resultant of adding vectors (2hat{i}) and (5hat{j}), you can follow a similar approach. Position the vectors with their tails at the origin, where (2hat{i}) is along the positive x-axis and (5hat{j}) is along the positive y-axis.

Draw a rectangle with these two vectors as sides. The diagonal of this rectangle represents the resultant vector (2hat{i} - 5hat{j}) in both magnitude and direction. Using vector addition properties, we can calculate the magnitude as:

(2hat{i} - 5hat{j} sqrt{2^2 (-5)^2} sqrt{4 25} sqrt{29})

The angle that the vector (2hat{i} - 5hat{j}) makes with the positive x-axis is:

(arctanleft(frac{-5}{2}right))

Conclusion

The resultant vector has a magnitude of (sqrt{29}) and points in a direction determined by the angle (theta arctanleft(frac{-5}{2}right)).