Question

If 'r'is position vector of a point, the div ris तो (A) 1 (B) 2 (C) 3 (E ( (D) 0 (​

Answer 1

Answer:

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Explanation:

The position vector of point R dividing the line segment joining two points P and Q in the ratio m:n is given by:

i. Internally:

m+n

m

Q

+n

P

ii. Externally:

m−n

m

Q

−n

P

Position vectors of P and Q are given as:

OP

=

i

^

+2

j

^

−

k

^

and

OQ

=−

i

^

+

j

^

+

k

^

(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,

OR

=

2+1

2(−

i

^

+

j

^

+

k

^

)+1(

i

^

+2

j

^

−

k

^

)

=

3

−2

i

^

+2

j

^

+

k

^

+(

i

^

+2

j

^

−

k

^

)

=

3

−

i

^

+4

j

^

+

k

^

=−

3

1

i

^

+

3

4

j

^

+

3

1

k

^

(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by,

OR

=

2−1

2(−

i

^

+

j

^

+

k

^

)−1(

i

^

+2

j

^

−

k

^

)

=(−2

i

^

+2

j

^

+2

k

^

)−(

i

^

+2

j

^

−

k

^

)

=−3

i

^

+3

k

^

Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

If $\vec{r}$is a position vector then divergence of $\vec{r}$

(A) 1

(B) 2

(C) 3

(D) 0

EVALUATION

The divergence of a continuously Differentiable vector point function $\vec{F}$ is denoted by $div \vec{F}$and defined as

$\displaystyle\nabla . \vec{F} = \bigg(\hat{i} \frac{ \partial \vec{F}}{ \partial x} + \hat{j} \frac{ \partial \vec{F}}{ \partial y} + \hat{k} \frac{ \partial \vec{F}}{ \partial z} \bigg)$

$Let \: \: \vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$

Hence the required divergence

$\displaystyle = \nabla . \vec{r}$

$\displaystyle = \bigg(\hat{i} \frac{ \partial \vec{r}}{ \partial x} + \hat{j} \frac{ \partial \vec{r}}{ \partial y} + \hat{k} \frac{ \partial \vec{r}}{ \partial z} \bigg)$

$\displaystyle = \bigg( \frac{ \partial x}{ \partial x} + \frac{ \partial y}{ \partial y} + \frac{ \partial z}{ \partial z} \bigg)$

$\displaystyle =1 + 1 + 1$

$\displaystyle =3$

FINAL ANSWER

Hence the correct option is (C) 3

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