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Find the unit vector in the same direction as the vector negative three 𝐢 plus five 𝐣.
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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.
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In this question, we have a vector with 𝐢 and 𝐣 components negative three and five.
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The magnitude of this vector is equal to the square root of negative three squared plus five squared.
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Negative three squared is equal to nine, and five squared is equal to 25.
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This means that the magnitude of vector 𝐕 is equal to root 34.
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The unit vector 𝐕 is therefore equal to one over root 34 multiplied by negative three, five.
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When multiplying any vector by a scalar, we multiply each individual component by the scalar.
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This gives us negative three over root 34, five over root 34.
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We can rationalize the denominator of one over root 34 by multiplying the numerator and denominator by root 34.
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This means that one over root 34 is equal to root 34 over 34.
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This is true of any radical.
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One over root 𝑎 is equal to root 𝑎 over 𝑎.
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We can therefore rewrite our two components as negative three root 34 over 34 and five root 34 over 34.
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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.