- vector D = 4 cm North
- vector J = 4.5 cm West
what is D+J?
In a more general sense, how can two 2D vectors that are perpendicular to each other be added?
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You really need to look at an introductory book on vectors because any answer we give on this site can only cover a tiny bit of the properties of vectors.
Having said that: you can add any vectors by thinking of them as a movement. For example vector D means "go 4cm North" and vector J means "go 4.5cm West". Adding the vectors then just means making the two movements ie D + J = "go 4cm North and 4.5cm West".
The sum D+J is the vector from the staring point to the end point shown by the dashed line. Using this method you can add any two vectors in any two directions.
This addition is exactly what Asdfsdjlka is doing in his answer. He's representing the vector by two numbers $(x, y)$ where $x$ means the direction East and $y$ means the direction North. Then D is (0, 4) i.e. zero cm East and 4 cm North and J is (-4.5, 0) i.e. -4.5 cm East and zero cm North. Representing vectors in this way is convenient for addition because for any two vectors $(x_1, y_1)$ and $(x_2, y_2)$ the sum of the two vectors is just $(x_1 + x_2, y_1 + y_2)$. This works whether the vectors are parallel, perpendicular or indeed at any angle. It also works for vectors in 3D where the vector has the form $(x, y, z)$.