Video: Expressing a Given Vector in Terms of Fundamental Unit Vectors

Given that 𝐀 = ⟨2, 0, 2⟩ and 𝐁 = ⟨0, 5, 9⟩, express the vector 𝐀𝐁 in terms of the unit vectors 𝐢, 𝐣, and 𝐤.

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Video Transcript

Given that vector 𝐀 is equal to two, zero, two and vector 𝐁 is equal to zero, five, nine, express the vector 𝐀𝐁 in terms of the unit vectors 𝐢, 𝐣, and 𝐤.

We begin by recalling that the vector 𝐀𝐁 is equal to the vector 𝐁 minus the vector 𝐀. In this question, we need to subtract the vector two, zero, two from the vector zero, five, nine. When subtracting vectors, we subtract the corresponding components. Zero minus two is equal to negative two, five minus zero is equal to five, and nine minus two is equal to seven. The vector 𝐀𝐁 has components negative two, five, and seven.

We recall that the unit vectors 𝐢 hat, 𝐣 hat, and 𝐤 hat are vectors of magnitude one in the positive 𝑥-, 𝑦-, and 𝑧-directions. We have worked out that the 𝑥-component of our vector is negative two, the 𝑦-component is five, and the 𝑧-component is seven. This means that the vector 𝐀𝐁 written in terms of its unit vectors 𝐢, 𝐣, and 𝐤 is equal to negative two 𝐢 plus five 𝐣 plus seven 𝐤.

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