# Video: Minimizing the Resultant Force on an Object

Two forces 𝐀 and 𝐁 act on an object. If 𝜃 is the angle between the directions of the two forces, for what value of 𝜃 is the resultant force minimum? [A] 180° [B] 45° [C] 0° [D] 60° [E] 90°

05:29

### Video Transcript

Two forces 𝐀 and 𝐁 act on an object. If 𝜃 is the angle between the directions of the two forces, for what value of 𝜃 is the resultant force minimum? a) 180 degrees, b) 45 degrees, c) zero degrees, d) 60 degrees, e) 90 degrees.

To begin answering the question, we can visualise each answer choice by drawing the object and the two corresponding forces acting on it. For answer choice a), our object has two forces acting on it that are 180 degrees apart from each other. This means that they are acting in opposite directions. Because the problem did not state the magnitude of the two forces, we have chosen to draw them at different lengths to represent a general case. The problem also did not state the direction of the forces, only the angle between them. We have chosen to draw vector 𝐀 to the right and vector 𝐁 to the left, ensuring that the angle between the two is 180 degrees as specified in the problem.

We have also chosen to draw the vectors different colours to help us when we add them together later. For choice b), our vectors are at 45 degrees to each other. We have chosen to draw vector 𝐀 to the right and vector 𝐁 at 45 degrees above vector 𝐀. For choice c), our vectors are at zero degrees to one another. This means that vectors 𝐀 and 𝐁 are pointing in the same direction. We have chosen to draw vector 𝐀 and 𝐁 pointing to the right. For answer choice d), the vectors are 60 degrees to one another. We have chosen to draw vector 𝐀 to the right and vector 𝐁 at 60 degrees above vector 𝐀. For choice e), our vectors are at 90 degrees to one another. We have chosen to draw vector 𝐀 to the right and vector 𝐁 at 90 degrees above vector 𝐀.

To determine which of the angles has a resultant force that is minimum, we must add our two vectors together. A quick way to add vectors without having to do any math is the tip-to-tail method. The tip-to-tail method is when we slide one vector until the tail of the vector is on the tip of the other vector. For each of our answer choices, we will slide vector 𝐁 over such that the tail of vector 𝐁 will be on the tip of vector 𝐀. For answer a), we start by drawing vector 𝐀 to the right. We draw vector 𝐁 starting at the tip of vector 𝐀 and pointing to the left. The resultant force will be pointing in the direction from where we began to where we ended.

In this case, our resultant started at the tail of 𝐀 and ended at the tip of 𝐁. As can be seen in our diagram, the resultant force is represented by a small arrow pointing to the right. For answer choice b), we once again begin by choosing to draw vector 𝐀 to the right. We then slide vector 𝐁 over until the tail of vector 𝐁 is on the tip of vector 𝐀. Our resultant force begins at the tail of vector 𝐀 and ends at the tip of vector 𝐁. As can be seen in the diagram we just drew, the size of resultant vector 𝑅 is larger than vector 𝐀, since it starts at the same place as the tail of 𝐀 and ends beyond the tip of 𝐀.

This is because vector 𝐁 has a horizontal component in the same direction as vector 𝐀. The component of vector 𝐁 adds to vector 𝐀 to make a larger resultant, where, in answer choice a), the resultant vector ends up being smaller than vector 𝐀 because vector 𝐁 subtracts from vector 𝐀 as they’re pointing in opposite directions. For answer choice c), we’ll once again begin by drawing vector 𝐀 to the right. We slide vector 𝐁 over until the tail of vector 𝐁 is on the tip of vector 𝐀. Our resultant vector will begin at the tail of vector 𝐀 and end at the tip of vector 𝐁. Because vector 𝐀 and vector 𝐁 are pointing in the same direction, they are added together to create a resultant vector that is the largest so far.

For answer choice d), we once again begin by drawing vector 𝐀 to the right. We slide vector 𝐁 over until the tail of vector 𝐁 is on the tip of vector 𝐀. The resultant vector starts at the tail of vector 𝐀 and ends at the tip of vector 𝐁. The resultant vector 𝑅 has a size that is larger than vector 𝐀. Similar to answer choice b), when we add the two vectors together, there is a horizontal component of vector 𝐁 in the same direction as vector 𝐀 that adds to make the resultant vector larger.

For the final answer choice e), we begin by drawing vector 𝐀 to the right. We slide vector 𝐁 over so the tail of vector 𝐁 is on the tip of vector 𝐀. The resultant vector begins at the tail of vector 𝐀 and ends at the tip of vector 𝐁. Because vector 𝐁 does not have a component in the same direction as vector 𝐀, we can see that our resultant vector 𝑅 is smaller than answer choices b), c), and d). To determine which angle gives us the minimum resultant force, we must compare our diagrams. We can see from the diagrams that answer choice a), 180 degrees, is the smallest resultant force.

Analysing the diagrams, we can see that answer choice a) is the only one that has vector 𝐁 being subtracted from vector 𝐀, yielding a smaller resultant vector. This was due to the orientation of the vectors compared to one another. Vectors that are oriented 180 degrees to one another will always produce a minimum resultant vector. Whereas vectors that are oriented zero degrees to one another will always yield a maximum resultant vector. As can be seen in answer choice c), where the two vectors are pointing in the same direction. Therefore, all of vector 𝐁 gets added to vector 𝐀 rather than just a component.