Question Video: Finding a Vector between Two Points Mathematics

Video Transcript

In the given figure, find vector 𝐀𝐁.

We begin by recalling that the vector between any two points, 𝐴 𝑥 sub one, 𝑦 sub one, 𝑧 sub one and 𝐵 𝑥 sub two, 𝑦 sub two, 𝑧 sub two, is given by vector 𝐀𝐁, which has components 𝑥 sub two minus 𝑥 sub one, 𝑦 sub two minus 𝑦 sub one, 𝑧 sub two minus 𝑧 sub one. In this question, we will begin by finding the coordinates of points 𝐴 and 𝐵 from the figure. From the origin, point 𝐴 is six centimeters in the positive 𝑥-direction. It is 12 centimeters in the positive 𝑦-direction and finally five centimeters in the positive 𝑧-direction. This means that point 𝐴 has coordinates six, 12, five. Repeating this process for point 𝐵, from the origin, we travel 12 centimeters in the positive 𝑦-direction and five centimeters in the positive 𝑧-direction. This means that point 𝐵 has coordinates zero, 12, five.

We can now calculate vector 𝐀𝐁 by subtracting the components of the corresponding position vectors. 𝐀𝐁 is equal to zero minus six, 12 minus 12, five minus five. This is equal to negative six, zero, zero. This is the vector 𝐀𝐁 from the diagram.

We could also have found this directly from the diagram. To travel from point 𝐴 to point 𝐵, we move six centimeters in the negative 𝑥-direction. This means that the 𝑥-component in our vector will be equal to negative six. As we don’t move in the 𝑦- or 𝑧-directions when traveling from point 𝐴 to point 𝐵, these will have components equal to zero. This confirms that vector 𝐀𝐁 is equal to negative six, zero, zero.