Chapter 9: Q. 54 (page 603)

Find a vector v whose magnitude is 3 and whose component in the i direction is equal to the component in the j direction.

Short Answer

Expert verified

Vector v is $\frac{3}{\sqrt{2}} \overset{⇀}{i} + \frac{3}{\sqrt{2}} \overset{⇀}{j}$

Step by step solution

Step. 1 Given

|v| = 3 and,

angle of the vector from both $\overset{⇀}{i} a n d \overset{⇀}{j}$ directions is same suppose that angle be $θ$ .

Step. 2 Vector representation

Now, As we know the general form of a vector is,

$\overset{⇀}{v} = r (\cos (θ) \overset{⇀}{i} + \sin (θ) \overset{⇀}{j})$

where, r is the magnitude of the vector i.e., distance from the origin and $θ$ is the angle of that vector from x-axis or $\overset{⇀}{i}$ direction.

Now, as magnitude in the question is given itself as 3. So,

role="math" localid="1646738176258" $r = 3$ .

So,

$\overset{⇀}{v} = 3 (\cos (θ) \overset{⇀}{i} + \sin (θ) \overset{⇀}{j})$

Step. 3 Final solution

Now, we have left with only $θ$ ,

So, as it is given that it makes equal angles with and we know these directions are mutually perpendicular thatswhy

$θ = 45 ° o r \frac{π}{4},$

Finally,

$\overset{⇀}{v} = 3 (\cos (\frac{π}{4}) \overset{⇀}{i} + \sin (\frac{π}{4}) \overset{⇀}{j}), \overset{⇀}{v} = 3 (\frac{\overset{⇀}{i}}{\sqrt{2}} + \frac{\overset{⇀}{j}}{\sqrt{2}}), \overset{⇀}{v} = \frac{3}{\sqrt{2}} \overset{⇀}{i} + \frac{3}{\sqrt{2}} \overset{⇀}{j} .$

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Find a vector v whose magnitude is 3 and whose component in the i direction is equal to the component in the j direction.

Short Answer

Step by step solution

Step. 1 Given

Step. 2 Vector representation

Step. 3 Final solution

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