# On occasion, the notation \vec A = [A, \theta] will be a shorthand notation for \vec A A \cos...

## Question:

On occasion, the notation {eq}\vec A = [A, \theta] {/eq} will be a shorthand notation for {eq}\vec A = A \cos \theta \hat i + A \sin \theta \hat j {/eq}. A vector {eq}\vec A {/eq} is added to {eq}\vec B = 6 \hat i+ 8 \hat j {/eq}. The resultant vector is in the positive {eq}x {/eq} direction and has a magnitude equal to {eq}\vec A {/eq} . What is the magnitude of {eq}\vec A {/eq}

a. 12.2

b. 11

c. 7.1

d. 8.3

e. 5.1

Description of motion in two dimensions necessitates the introduction of vectors. Vectors are directed quantities and are hence distinct from scalars. For example displacement of an object is a vector. Displacements in a plane cannot be added as simple numbers. The addition requires information about the relative directions of the component displacements.

Given that {eq}\vec A = A \cos \theta \hat i + A \sin \theta \hat j {/eq}

It is added to another vector {eq}\displaystyle {\vec B = 6 \hat i+ 8 \hat j } {/eq}

{eq}\displaystyle {\vec{A}+\vec{B}=( A \cos \theta+6) \hat i +(A \sin \theta+8) \hat j } {/eq}

Now, given that the resultant vector is in the positive x- direction.

This means that the y-component of the resultant vector is zero.

That is:

{eq}\displaystyle {A \sin \theta+8=0 \\ \text{or} \quad A \sin \theta=-8 \quad...............(1)} {/eq}

Then we can write:

{eq}\displaystyle {\vec{A}+\vec{B}=( A \cos \theta+6 ) \hat i } {/eq}

The magnitude of the resultant is given to the same as that of {eq}\displaystyle {\vec{A} } {/eq}

This gives :

{eq}\displaystyle {|\vec{A}+\vec{B}|=A \cos \theta+6=A \\ \text{or} \quad A \cos \theta=A-6 \quad...........(2) } {/eq}

Sqaring and adding equations (1) and (2) we get:

{eq}\displaystyle {A^2(\cos^2 \theta+\sin^2 \theta)=(A-6)^2+64 } {/eq}

{eq}\displaystyle {\text{or} \quad A^2=A^2-12A+100 } {/eq}

{eq}\displaystyle {\text{or} \quad A= \frac{100}{12}=8.3 } {/eq}

Hence Option (d) is the correct answer.