Question

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?

Short answer

Answer: (a) The directions of the two vectors are the same (0° between them) when the sum has a magnitude of 7.0. (b) The angle between the directions of the two vectors is either 120° or 240° when the sum has a magnitude of 5.0.

Step-by-step solution

Part (a)

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.

Part (b)

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 \(240^{\circ}\).

Part (c)

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 \(1\).