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Recognize a vector as a movement with length and direction, not as a fixed drawing on the page. Compare arrows that start in different places but represent the same displacement.
Use what you learned in the previous lesson to solve real-world problems.
Translate an arrow on a coordinate grid into a list like ⟨3, -2⟩ by counting horizontal and vertical change. Use each component as an instruction for how far to move along one axis.
Check what you understood with a short quiz.
Tell whether a vector lives in 2D, 3D, or a higher-dimensional setting by counting its components. Interpret each slot in the list as one measurement direction or feature.
Add two vectors by placing the tail of one at the tip of the other and reading the total movement. Connect this picture to combining two trips into one net displacement.
Add vectors in list form by adding matching components only. Check that ⟨a, b⟩ + ⟨c, d⟩ matches the same result you would get from the arrow picture.
Scale a vector by multiplying every component by the same number. Predict how positive, negative, zero, and fractional scalars change an arrow’s length and direction.
Use the zero vector as the “no change” vector for arrows, lists, and measurements. Reason through why adding it changes nothing and why scaling any vector by zero gives it.
Distinguish a position point like (4, 1) from a vector like ⟨4, 1⟩ by asking whether it names a location or a movement. See how a vector can point from the origin to a point without becoming the same kind of object.
Interpret vectors whose components are prices, forces, pixels, ratings, or features instead of map directions. Decide when adding or scaling those measurements makes real-world sense.
Combine scaled vectors such as 2u - 3v as a sequence of stretches, flips, and additions. Use the result as a first glimpse of how linear algebra builds new vectors from known ones.
Review this chapter with practice based on your mistakes.
