A vector in 2 dimensions (X and Y) can be decomposed into its X component and Y component. Each component is the magnitude, or length, of the vector in that dimension. So, in the picture, the vector A has an X component of 2 and a Y component of 3.

When we add vectors together, we add them component-wise. That is, we add the X components to get the X component of the resultant vector and then we add the Y components to get the Y component of the resultant vector. As an example, if we have vector A = [2,3]and vector B = [2,-1] then when we add them the resultant vector C will be [4,2]. That's because the X component of A is 2 and the X component of B is 2, so 2 + 2 = 4. Likewise, for the Y component of C: 3 + (-1) = 2. Geometrically, this is the same as putting the vectors head-to-tail. So, to find A + B, just move B so that its tail (the end without the arrow) is stuck to the head of A. Then you can draw a new vector from the tail of A to the head of B. That new vector is the resultant vector, C.

Given the image above, answer the questions below.

Vector subtraction is component-wise just like addition, but when we want to see vector subtraction geometrically it helps to think about it differently. In general we know that "A - B" is the same as "A + (-B)". Well, that holds for vectors as well. With vectors "-B" is the same magnitude, or length, as "B" but it points in the opposite direction. For example if B is the vector [1,3] then -B is the vector [-1,-3]. Then you can just use the same head-to-tail system you used to add vectors.

If you want to treat the vectors involved in the subtraction as "position vectors" then there's yet another way you can think about it! A position vector is a vector that describes a position in space. That is, if we translate (slide) the vector so that its tail is at the origin then its head points to a position which is the same as the X and Y components of the vector. In that situation "A - B" is like asking the question "What direction and how far do I need to travel from B to get to A?" If A and B are both position vectors then their tails are at the origin. Geometrically then, "A - B" is the vector you draw from the head of B to the head of A. Something to notice is that the resultant vector here is the same vector as it was in the head-to-tail method, it's just translated. Really, you can use either method but you might have to translate the result in order to answer the question you're actually asking.

Given the image above, answer the questions below.

You are in control of a small robot sent to survey a rift system and collect minerals on the newly discovered planet Kepler-42. The robot is equipped with a short range teleportation module that allows several smaller vectors to be input which will be added together in a series of jumps to reach a destination. Your mission is to find the series of vectors for teleportation that will allow the robot to traverse the rift safely.

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Drag vectors from the surrounding area and connect them end to end to guide the robot to its destination. (They'll snap to the yellow points). Some vectors will need to be negated (made negative) for the path to make sense. Just click on a vector and then click "Negate Last Clicked Vector" to negate that vector. All the vectors will be used and there are multiple configurations that will solve the problem.

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