Question

How do you calculate unit vectors?

Answer 1

Hint: In this question, we will first see the definition of a unit vector; also we will see the mathematical expression to find the unit vector. Further, we will use a problem to understand better. Here, by substituting the given values we get the required result. Also, we will discuss the basics of vector and unit vectors for our better understanding.
Formula used:
$a\hat i + b\hat j = \vec c$
$\hat v = \dfrac{v}{{\left| v \right|}}$

Complete answer:
As we know that a unit vector is defined as a vector of length one. A unit vector is also called a direction vector. The unit vector $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}\to
 {v} $having the same direction as a given (nonzero) vector v is defined by,
$\hat v = \dfrac{v}{{\left| v \right|}}$
Where $\left| v \right|$denotes the norm of v, is the unit vector in the same direction as the (finite) vector v .
Let us take a problem for our better understanding-
Problem- Find the unit vector which makes an angle of 60 degree with the vector i-k
Answer-
Let us take a vector which is given by:
$a\hat i + b\hat j = \vec c$
Also, we have:
$\vec c.(\hat i - \hat j) = \left| {\vec c} \right|\left| {\hat i - \hat j} \right|\cos {60^ \circ }$
Now, we substitute the values in above equation we get:
$(a\hat i + b\hat j).(\hat i - \hat j) = \left| {a\hat i + b\hat j} \right|\left| {\hat i - \hat j} \right|\cos {60^ \circ }$
Now, by substituting the given values in the above equation and solving, we get:
$(a - b) = \sqrt {{a^2} + {b^2}} \sqrt {{1^2} + {1^2}} \left( {\dfrac{1}{2}} \right)$
$\therefore (a - b) = \sqrt {{a^2} + {b^2}} \left( {\dfrac{1}{{\sqrt 2 }}} \right)$
Therefore, we get the required answer.

Additional information:
As we know that, ordinary quantities that have a magnitude but not direction are called scalars. Example: speed, time.
Also, a vector quantity is known as the quantity having magnitude and direction. Vector quantities must obey certain rules of combination.
These rules are:
1.VECTOR ADDITION: it is written symbolically as A + B = C. So, that it completes the triangle. Also, If A, B, and C are vectors, it should be possible to perform the same operation and achieve the same result i.e., C, in reverse order, B + A = C.
2.VECTOR MULTIPLICATION: This is the other rule of vector manipulation i.e., multiplication by a scalar- scalar multiplication. It is also termed as the dot product or inner product, and also known as the cross product.

Note:
We should remember that a vector has both magnitude and direction as well, whereas the scalar has only magnitude not the direction. Also, we should know that vectors can be used to find the angle of the resultant vector from its parent vectors.