Video: Expressing the Components of a Vector in Terms of the Standard Unit Vectors

Video Transcript

Express the vector 𝐙 with components negative five over two and negative 19 using the unit vectors 𝐢 and 𝐣.

So we’re given these two unit vectors 𝐢 and 𝐣. These are sometimes written with circumflex accents on top which look like hats. This emphasises the fact they are unit vectors, that is vectors whose magnitude is one. Of course, there are many unit vectors. There are unit vectors pointing in any direction you like. 𝐢 and 𝐣 are particular unit vectors; 𝐢 is the unit vector pointing in the 𝑥-direction, and 𝐣 is the unit vector pointing in the 𝑦-direction. 𝐢 has components one, zero and 𝐣 has components zero, one.

Any two-dimensional vector 𝐕 can be written in terms of 𝐢 and 𝐣. The vector 𝐕 is equal to the 𝑥-component of 𝑣 times 𝐢 plus the 𝑦-component of 𝑣 times 𝐣. So the vector 𝐙 is equal to the 𝑥-component of 𝐙, negative five over two, times 𝐢 plus the 𝑦-component of 𝐙, negative 19, times 𝐣.

Writing the plus negative 19𝐣 as minus 19𝐣, we get our final answer: 𝐙 is equal to negative five over two 𝐢 minus 19𝐣.

We can check this using the components of 𝐢 and 𝐣 and what we know about the scale of multiplication and subtraction of vectors. Negative five over two 𝐢 minus 19𝐣 is equal to negative five over two times the vector with components one, zero minus 19 times the vector with components zero, one. Here we’re using the component forms of 𝐢 and 𝐣. And of course, when multiplying a vector by a scaler, we just multiply the components of that vector by the scaler. So we get the vector with components negative five over two, zero minus the vector with components zero, 19. And subtracting one vector from another means subtracting the components of that vector from the other.

So we get the vector with components negative five over two, negative 19. And as desired, this is the vector 𝐙.