Question

Find the unit vector which makes an angle of 60 degree with the vector i-k

Answer 1

Hint: In this question, we will use the relation between the vector and its magnitude. Further, 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 the vector be:
$a\hat i + b\hat j = \vec c$
$\vec c.(\hat i - \hat j) = \left| {\vec c} \right|\left| {\hat i - \hat j} \right|\cos {60^ \circ }$
$(a\hat i + b\hat j).(\hat i - \hat j) = \left| {a\hat i + b\hat j} \right|\left| {\hat i - \hat j} \right|\cos {60^ \circ }$
Now, by substituting the given values in the above equation and solving, we get:
$(a - b) = \sqrt {{a^2} + {b^2}} \sqrt {{1^2} + {1^2}} \left( {\dfrac{1}{2}} \right)$
$(a - b) = \sqrt {{a^2} + {b^2}} \left( {\dfrac{1}{{\sqrt 2 }}} \right)$
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.
If we take example: we already know the basic difference of speed and velocity i.e., speed is the measure of how fast an object can travel, whereas velocity tells us the direction of this speed. Speed is a scalar quantity that means it has only magnitude, whereas velocity is a vector quantity that means it has both magnitude and direction. The S.I unit of velocity is meter per second (m/sec).

Note:
We should remember that vectors have 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.