Understanding the Magnitude and Direction of the Vector i j

In the context of vector algebra, the vector i j (where i and j are unit vectors along the x-axis and y-axis respectively) is a fundamental concept. This article will explain how to determine the magnitude and direction of the vector, along with a step-by-step breakdown of the calculations and the geometric interpretation.

Magnitude of the Vector i j

The magnitude of a vector is a scalar quantity that represents the length or size of the vector. For the vector mathbf{v} mathbf{i} mathbf{j}, where mathbf{i} and mathbf{j} are unit vectors, the magnitude can be calculated using the formula:

mathbf{v} sqrt{a^2 b^2}

Given that both a and b are 1 (since mathbf{i} and mathbf{j} are unit vectors), the calculation of the magnitude is as follows:

mathbf{v} sqrt{1^2 1^2} sqrt{1 1} sqrt{2}

The magnitude of the vector i j is sqrt{2} approx 1.41.

Direction of the Vector i j

The direction of a vector can be determined by the angle it makes with the positive x-axis. This angle, denoted as theta, can be calculated using the tangent function:

tan theta frac{b}{a}

For the vector mathbf{v} mathbf{i} mathbf{j}, where a 1 and b 1 (both unit vector components), the angle is:

tan theta frac{1}{1} 1

To find the angle theta, we take the inverse tangent (arctangent) of 1, giving us:

theta tan^{-1} (1) 45^circ

The vector i j is thus directed at an angle of 45 degrees from the positive x-axis.

Geometric Interpretation

The vector i j can be visualized as the diagonal of a unit square in the Cartesian coordinate system. When two unit vectors (mathbf{i}) and (mathbf{j}) are added, they form a parallelogram, which in the case of these unit vectors, becomes a square. The resultant vector, which is the diagonal of this square, has a magnitude of (sqrt{2}) and is inclined at 45 degrees to the positive x-axis.

This interpretation is consistent with the mathematical calculation, as showed that the length of the diagonal of a unit square is (sqrt{2}).

In a more general context, the Cartesian representation of a vector (mathbf{A} A_xmathbf{i} A_ymathbf{j} A_zmathbf{k}) has magnitude:

A sqrt{A_x^2 A_y^2 A_z^2}

And the direction with respect to the x-axis is given by:

theta cos^{-1} (frac{A_x}{A})

For the vector i j in our problem, the components are (A_x 1) and (A_y 1), with no z-component. Using the above formula for magnitude and the angle calculation, we confirm the same results as derived earlier.

Conclusion

Understanding the magnitude and direction of vectors is crucial in various fields of physics and engineering. The vector i j has a magnitude of (sqrt{2}) and is directed at 45 degrees to the positive x-axis. This concept can be extended to more complex vector operations using the principles of vector addition and scalar multiplication.

Keywords

vector magnitude, vector direction, unit vector