Question Video: Writing a Vector Represented Graphically in Terms of Fundamental Unit Vectors Mathematics • First Year of Secondary School

Video Transcript

The given figure shows a vector 𝐀 in a plane. Express this vector in terms of the unit vectors 𝐢 and 𝐣.

We’re asked to express the vector shown in the graph in terms of the fundamental unit vectors 𝐢 and 𝐣. So let’s remind ourselves what these look like in the Cartesian plane. 𝐢 is a horizontal vector and 𝐣 is a vertical vector, and they both start at a given origin traveling in the positive 𝑥- and 𝑦-directions, respectively, with length equal to one.

Now, to express the vector 𝐀 in terms of 𝐢 and 𝐣, we need to consider its 𝑥- and 𝑦-components separately. So let’s begin with the 𝑥-component. From the graph, we see that the 𝑥-component is equal to negative three. Using the horizontal unit vector 𝐢, which is going in the positive 𝑥-direction, we flip this unit vector so it becomes negative, as it’s now pointing in the negative 𝑥-direction. And to reach 𝑥 is negative three, we’re adding three copies of negative 𝐢. This tells us that the 𝑥-component of 𝐀 in terms of the fundamental unit vector 𝐢 is negative three 𝐢.

Next, considering the 𝑦-component of 𝐀, we see that this is positive two. And we can produce this by adding two copies of 𝐣. Hence, the 𝑦-component of vector 𝐀 in terms of the fundamental unit vector 𝐣 is two 𝐣.

So now adding the two components together, where by convention we write the 𝑥-component first, we have 𝐀 equals negative three 𝐢 plus two 𝐣.