Question Video: Expressing a Given Vector in Cartesian Coordinates Mathematics • 12th Grade

Given that 𝐀 = 9𝐢 + 3𝐣 + 7𝐤, write vector 𝐀 in Cartesian coordinates.

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Video Transcript

Given that vector 𝐀 is equal to nine 𝐢 plus three 𝐣 plus seven 𝐤, write vector 𝐀 in Cartesian coordinates.

Vector 𝐀 is currently written in terms of the fundamental unit vectors 𝐢 hat, 𝐣 hat, and 𝐤 hat. These can be shown on the three-dimensional coordinate plane as follows. The unit vector 𝐢 hat corresponds to one unit in the 𝑥-direction. The unit vector 𝐣 hat corresponds to one unit in the positive 𝑦-direction. Finally, the unit vector 𝐤 hat corresponds to one unit in the positive 𝑧-direction.

We can write any vector given in this form in terms of its Cartesian coordinates. This is written 𝑥, 𝑦, 𝑧 as shown. Vector 𝐀 is equal to nine 𝐢 plus three 𝐣 plus seven 𝐤. Therefore, our 𝑥-value is nine, the 𝑦-value is equal to three, and the 𝑧 or 𝑧-value is equal to seven. The vector 𝐀 nine 𝐢 plus three 𝐣 plus seven 𝐤 written in Cartesian coordinates is nine, three, seven.