(a) What's the magnitude of \(\hat{\imath}+\hat{\jmath} ?\) (b) What angle does it make with the \(x\) -axis?

Starting with the basics, the Pythagorean Theorem is a fundamental element in mathematics, especially when dealing with right triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Laid out simply, this can be written as the equation: \[a^2 + b^2 = c^2\]
where \(a\) and \(b\) are the lengths of the legs of the triangle and \(c\) is the length of the hypotenuse.

When applying the Pythagorean Theorem to calculate vector magnitude, we treat the vector's components along the x and y axes as the legs of a right triangle. The vector itself represents the hypotenuse. Hence, if the vector's components are \(1\) along both the x and y axes—as in our example of \( \hat{\imath} + \hat{\jmath} \)—the magnitude is calculated using the formula: \[\lVert v \rVert = \sqrt{x^2 + y^2} = \sqrt{1^2 + 1^2} = \sqrt{2}\].

This concept not only aids in finding vector lengths but also serves as a foundational tool used across numerous areas of STEM fields.

The inverse tangent function, often represented by \(atan\) or \(tan^{-1}\), is crucial when it comes to determining angles, particularly in triangle geometry and trigonometry. This function is the reverse process of finding the tangent of an angle, which means that if \(tan(\Theta) = \frac{y}{x}\), then \(\Theta = atan(\frac{y}{x})\).

When calculating angles in vectors, the inverse tangent function helps us determine the angle that a vector makes with the horizontal or x-axis. Take our example; the vector has equal lengths along both axes, giving us an angle of: \[\Theta = atan(\frac{y}{x}) = atan(\frac{1}{1}) = atan(1)\].

This calculation yields an angle of \(45^\circ\) or \(\frac{\pi}{4}\) radians, which is exactly the case for our vector. It's important to note that the angle found using the inverse tangent function tells us the angle within the right triangle formed by the vector components, not necessarily the absolute angle on a plane, which is where quadrant considerations come in.

Vector components are essentially the building blocks of a vector. They represent how much of the vector lies along each axis in a coordinate system. For instance, a two-dimensional vector has two components: one along the x-axis, called the horizontal component, and one along the y-axis, known as the vertical component.

In mathematical terms, for a vector \(v\) represented as \(v = a\hat{\imath} + b\hat{\jmath}\), \(a\) and \(b\) signify the magnitudes of the vector's components along the x and y axes, respectively. Here, \(\hat{\imath}\) and \(\hat{\jmath}\) are unit vectors that give the direction of the components on the coordinate plane.

The vector components help in breaking down the action of the vector into manageable parts, making it simpler to analyze and perform various operations like addition, subtraction, and even finding perpendicular components. In our exercise, the components are both 1, which infers that the vector is equally aligned along both axes, simplifying computations and visual representation on a graph. They serve as a practical tool not only in physics for resolving forces but also in computer graphics, engineering, and several other fields requiring spatial analysis.