Question Video: Find a Unit Vector in Terms of the Original Vector Mathematics

Find the unit vector in the same direction as the vector 𝐢 − 𝐣 − 𝐤.

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Video Transcript

Find the unit vector in the same direction as the vector 𝐢 minus 𝐣 minus 𝐤.

The unit vector denoted 𝐕 hat is equal to one over the magnitude of 𝐕 multiplied by the vector 𝐕. We also recall that the magnitude of vector 𝐕 is equal to the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared, where 𝑥, 𝑦, and 𝑧 are the components of 𝐢, 𝐣, and 𝐤, respectively. The magnitude of our vector is therefore equal to the square root of one squared plus negative one squared plus negative one squared. As both one and negative one squared are equal to one, this is equal to the square root of three.

The unit vector of 𝐢 minus 𝐣 minus 𝐤 is therefore equal to one over root three multiplied by 𝐢 minus 𝐣 minus 𝐤. We know that when we have a radical on the denominator, we can rationalize this such that one over the square root of 𝑎 is equal to the square root of 𝑎 over 𝑎. This means that one over the square root of three is the same as the square root of three over three. The unit vector, written in its simplest form, is therefore equal to root three over three multiplied by 𝐢 minus 𝐣 minus 𝐤.

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