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

Given that 𝐴 = (9, 3, 2), express vector 𝐀𝐎 in terms of the unit vectors 𝐢, 𝐣, and 𝐤.

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Video Transcript

Given that the point 𝐴 has coordinates nine, three, two, express vector 𝐀𝐎 in terms of the unit vectors 𝐢, 𝐣, and 𝐤.

We begin by recalling that the vector 𝐎𝐀 in three-dimensional space will have 𝑥-, 𝑦-, and 𝑧-components equal to the 𝑥-, 𝑦-, and 𝑧-coordinates of point 𝐴. This means that, in this question, vector 𝐎𝐀 is equal to nine, three, two. Recalling that the unit vectors 𝐢 hat, 𝐣 hat, and 𝐤 hat are vectors of magnitude one in the positive 𝑥-, 𝑦-, and 𝑧-directions, this vector 𝐎𝐀 can be rewritten as nine 𝐢 plus three 𝐣 plus two 𝐤. The vector 𝐀𝐎 will have the same magnitude as the vector 𝐎𝐀 but will act in the opposite direction. Therefore, the vector 𝐀𝐎 is equal to the negative of the vector 𝐎𝐀.

Vector 𝐀𝐎 is therefore equal to negative nine 𝐢 minus three 𝐣 minus two 𝐤. To travel from the point 𝐴 to the origin 𝑂, we move negative nine units in the 𝑥-direction, negative three units in the 𝑦-direction, and negative two units in the 𝑧-direction.

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