Answer

Verified

348.9k+ views

**Hint:**In order to solve this question, we are first going to consider the two vectors, and consider the magnitudes for them. Now, we will consider the parallelogram with the two vectors \[A\] and \[B\], and the resultant is given by the diagonal of the parallelogram. Then, the magnitude and direction of the resultant are found.

**Formula used:**

The Pythagoras theorem is given by,

\[O{C^2} = C{D^2} + D{O^2}\]

Where, \[OC\] is the hypotenuse, \[CD\] and \[DO\] are base and perpendicular.

**Complete answer:**

It is given that there are two vectors \[A\] and \[B\], such that their magnitudes are

$ \left| A \right| = a \\

\left| B \right| = b \\ $

The angle between them is \[\theta \]

So, if the resultant of these two vectors is taken as \[C\]

Then,

\[A + B = C\]

Now, to find the magnitude and the direction, consider the parallelogram \[OABC\] with \[D\] as the foot of the perpendicular as shown in the figure

Now, in triangle \[OCD\], if we apply the Pythagoras theorem,

\[O{C^2} = C{D^2} + D{O^2}\]

Solving this further, we get

\[OC = \sqrt {O{A^2} + A{D^2} + 2 \cdot OA \cdot AD + C{D^2}} \]

Putting the values,

\[OC = \sqrt {{a^2} + {b^2} + 2 \cdot a \cdot b\cos \alpha } \]

Where,\[AD = AC\cos \alpha = b\cos \alpha \]

Thus, the magnitude of the resultant vector is

\[c = \sqrt {{a^2} + {b^2} + 2ab\cos \alpha } \]

Now, finding the direction, we get

Given that \[C\] makes an angle \[\theta \] with \[A\],

\[\theta = {\tan ^{ - 1}}\left( {\dfrac{{b\sin \alpha }}{{a + b\cos \alpha }}} \right)\]

Hence, the magnitude and direction of the resultant are given by

$ c = \sqrt {{a^2} + {b^2} + 2ab\cos \alpha } \\

\theta = {\tan ^{ - 1}}\left( {\dfrac{{b\sin \alpha }}{{a + b\cos \alpha }}} \right) \\ $

**Note:**It is important to note that the triangular law of addition has been applied in this question, for the vectors \[A\] and \[B\],

\[A + B = C\]

According to the triangle law of vector addition:

Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Guru Purnima speech in English in 100 words class 7 english CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Select the word that is correctly spelled a Twelveth class 10 english CBSE