Question Video: Finding the Coordinates of a Vector Mathematics

If 𝐀𝐁 = 𝐢 − 2𝐣 and 𝐁 ⟨4, 3⟩, then the coordinates of vector 𝐀 are _.

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Video Transcript

If vector 𝐀𝐁 is equal to 𝐢 minus two 𝐣 and vector 𝐁 equals four, three, then the coordinates of vector 𝐀 are blank.

We begin by recalling that vector 𝐀𝐁 is equal to vector 𝐁 minus vector 𝐀. We also know that any vector 𝐕 written in terms of unit vectors 𝐢 and 𝐣 such that 𝐕 is equal to 𝑥𝐢 plus 𝑦𝐣 can be rewritten such that vector 𝐕 has components 𝑥 and 𝑦. The vector 𝐢 minus two 𝐣 can, therefore, be rewritten in terms of its components as one, negative two. This must be equal to the vector four, three minus the vector 𝑥, 𝑦 where 𝑥 and 𝑦 are the components of vector 𝐀.

We recall that when adding and subtracting vectors, we simply add or subtract the corresponding components. When considering the 𝑥-components, we have the equation one is equal to four minus 𝑥. Adding 𝑥 and subtracting one from both sides of this equation gives us 𝑥 is equal to four minus one. This gives us a value of 𝑥 equal to three. Repeating this for our 𝑦-components, we have the equation negative two is equal to three minus 𝑦. We can then add 𝑦 and two to both sides of this equation such that 𝑦 is equal to three plus two. This gives us a 𝑦-component equal to five.

Vector 𝐀, therefore, has coordinates three, five.

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