Video: Finding Unit Vectors in the Same Direction as a Given Vector

Find the unit vector in the same direction as the vector −3𝐢 + 5𝐣.

02:16

Video Transcript

Find the unit vector in the same direction as the vector negative three 𝐢 plus five 𝐣.

We know that the unit vector 𝐕 hat is equal to one over the magnitude of vector 𝐕 multiplied by vector 𝐕, where the magnitude of a two-dimensional vector with components 𝑎 and 𝑏 is equal to the square root of 𝑎 squared plus 𝑏 squared.

In this question, we have a vector with 𝐢 and 𝐣 components negative three and five. The magnitude of this vector is equal to the square root of negative three squared plus five squared. Negative three squared is equal to nine, and five squared is equal to 25. This means that the magnitude of vector 𝐕 is equal to root 34. The unit vector 𝐕 is therefore equal to one over root 34 multiplied by negative three, five.

When multiplying any vector by a scalar, we multiply each individual component by the scalar. This gives us negative three over root 34, five over root 34. We can rationalize the denominator of one over root 34 by multiplying the numerator and denominator by root 34. This means that one over root 34 is equal to root 34 over 34. This is true of any radical. One over root 𝑎 is equal to root 𝑎 over 𝑎.

We can therefore rewrite our two components as negative three root 34 over 34 and five root 34 over 34. Rewriting this in terms of 𝐢 and 𝐣, the unit vector in the same direction as the vector negative three 𝐢 plus five 𝐣 is equal to negative three root 34 over 34 𝐢 plus five root 34 over 34 𝐣.

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