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.