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.
