Two vectors have magnitudes 3.0 and \(4.0 .\) How are the directions of the two vectors related if (a) the sum has magnitude \(7.0,\) or \((\mathrm{b})\) if the sum has magnitude \(5.0 ?\) (c) What relationship between the directions gives the smallest magnitude sum and what is this magnitude?

In this part, the sum of the two vectors has a magnitude of \(7.0\). Let's denote vector \(A\) with magnitude 3.0 and vector \(B\) with magnitude 4.0. When vector \(A\) is added to vector \(B\), the resulting vector \(R\) has a magnitude of \(7.0\). The sum of two vectors can be represented as: \(R = A + B\) To find the relationship between the directions of the two vectors, we can use the magnitude formula for the sum of two vectors: \(|R|^2 = |A|^2 + |B|^2 + 2|A||B|\cos{(\theta)}\) Here, \(\theta\) is the angle between vectors \(A\) and \(B\). Substituting the given values, we have: \(7^2 = 3^2 + 4^2 + 2(3)(4)\cos{(\theta)}\) Solving for \(\cos{(\theta)}\), we get: \(\cos{(\theta)} = 1\) This implies that the angle \(\theta = 0^{\circ}\). The directions of the two vectors are the same in this case.

In this part, the sum of the two vectors has a magnitude of \(5.0\). Using the same approach as in part (a), we can use the magnitude formula for the sum of two vectors and substitute the given values: \(5^2 = 3^2 + 4^2 + 2(3)(4)\cos{(\theta)}\) Solving for \(\cos{(\theta)}\), we get: \(\cos{(\theta)} = -\frac{1}{2}\) This implies that the angle between the two vectors is either \(\theta = 120^{\circ}\) or \(240^{\circ}\).

To find the relationship between the directions that gives the smallest magnitude sum, we need to find the smallest possible value of \(|R|^2\). Taking the derivative of the magnitude formula with respect to \(\theta\) and setting it to zero, we get the condition for the smallest magnitude sum: \(\frac{d|R|^2}{d\theta} = -2|A||B|\sin{(\theta)} = 0\) This implies that \(\sin{(\theta)} = 0\), which means the angle between the two vectors is either \(0^{\circ}\) or \(180^{\circ}\). However, we have already analyzed the case when the angle is 0, which resulted in the sum having a magnitude of \(7.0\). So the smallest possible magnitude occurs when the angle between the vectors is \(180^{\circ}\). At this angle, the sum has the smallest possible magnitude, and we can calculate this magnitude as follows: \(|R|^2 = 3^2 + 4^2 - 2(3)(4) = 1\) Taking the square root, we get the smallest possible magnitude of the sum: \(|R| = 1\) So, the relationship between the directions that gives the smallest magnitude sum is when the angle between them is \(180^{\circ}\), and the smallest magnitude sum is \(1\).