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Question: Two vectors \( \overrightarrow a \) and \( \overrightarrow b \) have equal magnitudes of \( 12 \) un...

Two vectors a→\overrightarrow a and b→\overrightarrow b have equal magnitudes of 1212 units. These angles are making angles 30∘{30^ \circ } and 120∘{120^ \circ } with the x axis respectively. Their sum is r→\overrightarrow r . Find the x and y components of r→\overrightarrow r :

Explanation

Solution

Vector quantity is a quantity that has both magnitude and direction. A vector quantity has two characteristics, a magnitude and a direction. When comparing two vector quantities of the same type, we have to compare both the magnitude and the direction.

Complete step by step solution:
A vector that is directed in two dimensions can be considered to be having an influence in two different directions. This means that it can be thought to have two different parts. Each part of the two-dimensional vector is called a component. The components of a vector helps to depict the influence of that vector in a particular direction. The combined influence of both these components is equal to the influence of the two-dimensional single vectors. The single two-dimensional vector can be replaced by the two vector components.
As a→\overrightarrow a makes an angle of 30∘{30^ \circ } with the x- axis,
a→=12cos⁡30∘i^+12sin⁡30∘j^\overrightarrow a = 12\cos {30^ \circ }\widehat i + 12\sin {30^ \circ }\widehat j
a=63i^+6j^a = 6\sqrt 3 \widehat i + 6\widehat j
Similarly, as b→\overrightarrow b makes an angle of 120∘{120^ \circ } with x- axis,
b→=12cos⁡120∘i^+12sin⁡120∘j^\overrightarrow b = 12\cos {120^ \circ }\widehat i + 12\sin {120^ \circ }\widehat j
b→=−6i^+63j^\overrightarrow b = - 6\widehat i + 6\sqrt 3 \widehat j
Now, r→=a→+b→\overrightarrow r = \overrightarrow a + \overrightarrow b
On putting the required values in the above equation, we get,
r=(63i^+6j^)+(−6i^+63j^)r = (6\sqrt 3 \widehat i + 6\widehat j) + ( - 6\widehat i + 6\sqrt 3 \widehat j) 6(3+1)6(\sqrt 3 + 1)
On taking the x component and y component separately, we get,
r=6(3−1)i^+6(3+1)j^r = 6(\sqrt 3 - 1)\widehat i + 6(\sqrt 3 + 1)\widehat j
So the x component is 6(3−1)6(\sqrt 3 - 1).
And the y component is 6(3+1)6(\sqrt 3 + 1).

Note:
Most commonly in physics, vectors are used to represent displacement, velocity, and acceleration. Vectors are a combination of magnitude and direction, and are drawn as arrows. The length represents the magnitude and the direction of that quantity is the direction in which the vector is pointing.