Two vectors \(\vec{A}\) and \(\vec{B}\) have the same magnitude \(A\) and are at right angles. Find the magnitudes of (a) \(\vec{A}+2 \vec{B}\) and \((b) 3 \vec{A}-\vec{B}\).

When dealing with vectors, understanding the Pythagorean theorem is essential, especially for vectors at right angles to one another. The principle remains the same as in classic geometry: in a right-angle triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

For vectors, think of each side of the triangle as a vector and the hypotenuse as the resulting vector from their addition or subtraction. For example, when you have two perpendicular vectors \(\vec{A}\) and \(\vec{B}\), and you need to calculate the magnitude of their sum \(\vec{A}+\vec{B}\), the theorem tells us that \(\left|\vec{A}+\vec{B}\right| = \sqrt{\left|\vec{A}\right|^2 + \left|\vec{B}\right|^2}\).

Applying this to an exercise where we need to find the magnitude of vector \(\vec{A}+2\vec{B}\), given that \(\vec{A}\) and \(\vec{B}\) are perpendicular and have the same magnitude \(A\), we would use the Pythagorean theorem to arrive at the solution \(A\sqrt{5}\).

Vector addition and subtraction are operations that combine or differentiate vectors in terms of direction and magnitude. In a two-dimensional space, these operations can often be visualized as moving along a path in different directions.

For addition, you place the tail of one vector at the head of another and draw a vector from the tail of the first vector to the head of the second; this is known as the 'head-to-tail' method. Subtraction can be thought of as adding a negative vector; this means you flip the direction of the vector you're subtracting and then add as usual.

Using the example from our exercise, to compute \(3\vec{A}-\vec{B}\), one might visualize moving along \(3\vec{A}\) and then in the opposite direction of \(\vec{B}\). This visual representation aids in understanding the right triangle formation, which is then used with the Pythagorean theorem to find the result.

The magnitude of a vector is a measure of its length or size, irrespective of its direction, and it's always a non-negative value. In two dimensions, the magnitude can be calculated using the formula \(\left|\vec{V}\right| = \sqrt{V_x^2 + V_y^2}\), where \(V_x\) and \(V_y\) are the components of the vector along the x and y axes, respectively.

In the context of our exercise, the magnitude of a resultant vector (like \(\vec{A}+2\vec{B}\) or \(3\vec{A}-\vec{B}\)) can be determined by first resolving the vectors into their components, then applying the Pythagorean theorem. This provides a clear, step-by-step method for students to not only solve vector problems but also to understand the underlying principles governing vector magnitude.