Question Video: Expressing a Given Vector in Terms of Fundamental Unit Vectors | Nagwa

Video Transcript

Given that point 𝐴 is equal to two, three, point 𝐵 is equal to five, nine, express the vector 𝐀𝐁 in terms of the unit vectors 𝐢 and 𝐣.

We will begin by sketching the two-dimensional coordinate plane and plotting the points 𝐴 and 𝐵. Point 𝐴 has coordinates two, three, and point 𝐵 has coordinates five, nine. We recall that the unit vectors 𝐢 and 𝐣 are vectors of magnitude one in the positive 𝑥- and 𝑦-direction, respectively. We are asked to calculate the vector 𝐀𝐁.

One way of doing this is to subtract the corresponding coordinates of point 𝐴 from point 𝐵. The vector 𝐀𝐁 will have components five minus two and nine minus three. This is equal to three, six. And written in terms of the unit vectors 𝐢 hat and 𝐣 hat, vector 𝐀𝐁 is equal to three 𝐢 plus six 𝐣.

An alternative method here would be to recall that the vector 𝐀𝐁 is equal to the vector 𝐎𝐁 minus the vector 𝐎𝐀. The vector 𝐎𝐁 is equal to five 𝐢 plus nine 𝐣 as we move five units in the 𝑥-direction and nine units in the 𝑦-direction to get from point 𝑂, the origin, to point 𝐵. The vector 𝐎𝐀 is equal to two 𝐢 plus three 𝐣 as point 𝐴 has coordinates two, three. Subtracting vector 𝐎𝐀 from vector 𝐎𝐁, we once again find that vector 𝐀𝐁 is equal to three 𝐢 plus six 𝐣.