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Question: Find the component of \(\vec r\) in the direction of \(\vec a\) :- (A) \(\dfrac{{\left( {\vec r \c...

Find the component of r⃗\vec r in the direction of a⃗\vec a :-
(A) (r⃗⋅a⃗)a⃗a2\dfrac{{\left( {\vec r \cdot \vec a} \right)\vec a}}{{{a^2}}}
(B) (r⃗⋅a⃗)a⃗a\dfrac{{\left( {\vec r \cdot \vec a} \right)\vec a}}{a}
(C) (r⃗×a⃗)a⃗a2\dfrac{{\left( {\vec r \times \vec a} \right)\vec a}}{{{a^2}}}
(D) None of the above.

Explanation

Solution

The component can be found by using the scalar product of vectors. We need to compute the value of the unit vector to find the solution.

Complete Step by Step Solution: We know that the location of the points on the coordinate plane can be represented using the ordered pair such as (x,y)\left( {x,y} \right).
A vector of nn dimensions is an ordered collection of nn elements called components.
There are two components of a vector in the x−yx - y plane are the horizontal component and vertical component. Breaking a vector into its xx and yy components in the vector space is the most common way for solving vectors.
It has been given that there are two vectors r⃗\vec r and a⃗\vec a.
Let the desired component be p⃗\vec p. Since p⃗\vec p is in the same direction as a⃗\vec a, it must be a multiple of a⃗\vec a. This means that there is a scalar, i.e., a real number ccsuch that, p⃗=c⋅a⃗\vec p = c \cdot \vec a.
The component of r⃗\vec r along a⃗\vec a is the distance along a⃗\vec a obtained by dropping down a perpendicular line from r⃗\vec r.
The component of r⃗\vec r along a⃗\vec a is compar⃗=r⃗⋅a⃗∣a∣com{p_a}\vec r = \dfrac{{\vec r \cdot \vec a}}{{\left| a \right|}}
Simplifying, c=r⃗⋅a⃗aa^c = \dfrac{{\vec r \cdot \vec a}}{a}\hat a.
Since, a^=a⃗a\hat a = \dfrac{{\vec a}}{a}, c=r⃗−a⃗a2⋅a⃗c = \dfrac{{\vec r - \vec a}}{{{a^2}}} \cdot \vec a.

Hence the answer is Option A.

Note: Vector is an object which has magnitude and direction both. Magnitude defines the size of the vector. It is represented by a line with an arrow, where the length of the line is the magnitude of the vector and the arrow shows the direction. The angle at which the vector is inclined shows the direction of the vector. The starting point of a vector is called the Tail and its ending point with an arrow is called the Head of the vector.