Question

How do you find the unit vector having the same direction as the vector \[\overrightarrow{u}=3i-4j\] ?

Answer 1

Hint: As we can see that we are given a vector \[\overrightarrow{u}=3i-4j\] and we have to find a unit vector in the direction of this vector. Here we have to find a vector with magnitude 1. therefore, we will divide the given vector by its magnitude. So, we have to find the magnitude of the given vector using the formula \[|v|=\sqrt{{{a}^{2}}+{{b}^{2}}}\].

Complete step by step answer:
The above question belongs to the concept of unit vector in vector algebra. A unit vector, in a normal vector space is a vector whose length or magnitude is equal to 1. We usually denote a unit vector with a circumflex. It is also known as direction vector. A unit vector is commonly used to indicate direction with a scalar coefficient which is providing the magnitude. A vector can be decomposed into a sum of unit vector and scalar coefficients. A unit vector is equal to the ratio of a given vector and its magnitude.
In the question we have to find the unit vector having the same direction as of the vector
\[\overrightarrow{u}=3i-4j\] Therefore, we will divide the given vector with its magnitude. To find the magnitude of the given vector we will use the formula
\[|v|=\sqrt{{{a}^{2}}+{{b}^{2}}}\]
\[\begin{align}
  & \overrightarrow{u}=3i-4j \\
 & \Rightarrow |v|=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
 & \Rightarrow |u|=\sqrt{{{3}^{2}}+{{4}^{2}}}=\sqrt{25} \\
 & \Rightarrow |u|=5 \\
\end{align}\]
Therefore, the magnitude of the given vector is 5.
Now, we have to divide the given vector by 5.

Therefore, the required unit vector is
\[\dfrac{3}{5}i-\dfrac{4}{5}j\].

Note:While solving the above question keep in mind the definition of unit vector. Carefully use the notations in the above question for vectors. Sometimes in the question we are asked to find the unit vector in the opposite direction, in that case we have to follow the same procedure but at the end signs have to be reversed.