Final answer:
The angle between the diagonal of a unit square and one of the coordinate axes in R2 is 45 degrees or π/4 radians. This can be determined using either the Pythagorean theorem or trigonometric identities, specifically the inverse tangent function, given the equal sides of the unit square form a right triangle with the diagonal.
Step-by-step explanation:
The question asks to find the angle between the diagonal of a unit square and one of the coordinate axes in R2. Considering a unit square in the Cartesian coordinate system, the diagonal forms a right triangle with sides of equal length that are aligned with the x-axis and y-axis since the sides of the square are of length 1. Therefore, the diagonal being the hypotenuse has a magnitude calculated using the Pythagorean theorem. As both sides along the x-axis and y-axis are equal, we can determine that the angle formed between the diagonal and each axis is 45 degrees or π/4 radians.
The angle can alternatively be found using trigonometric identities. If we define the angle θ as the angle formed by the diagonal with one of the axes, we can use the inverse tangent function to find the angle since the opposite and adjacent sides of the right triangle formed by the diagonal are equal. The inverse tangent of 1 (the ratio of the y-component to the x-component of the diagonal) yields an angle of 45 degrees or π/4 radians.
When determining the direction of a vector using a protractor, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since we are working with the diagonal of a unit square, this measurement results in an angle of 45 degrees with either the horizontal (x-axis) or vertical (y-axis) depending on which side of the axis the diagonal lies.