Which of the following is independent of the choice of co-ordinate system

  • A

    $\vec P + \vec Q + \vec R$

  • B

    $({P_x} + {Q_x} + {R_x})\hat i$

  • C

    ${P_x}\hat i + {Q_y}\hat j + {R_z}\hat k$

  • D

    None of these

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Given that $\vec A\, + \,\vec B\, = \,\vec C\,.$  If  $\left| {\vec A} \right|\, = \,4,\,\,\left| {\vec B} \right|\, = \,5\,\,$ and $\left| {\vec C} \right|\, =\,\sqrt {61}$ the angle between $\vec A\,\,$ and $\vec B$ is ....... $^o$

The magnitude of a given vector with end points $ (4, -4, 0)$ and $(-2, -2, 0)$ must be

Assertion $A$ : If $A, B, C, D$ are four points on a semi-circular arc with centre at $'O'$ such that $|\overrightarrow{{AB}}|=|\overrightarrow{{BC}}|=|\overrightarrow{{CD}}|$, then $\overrightarrow{{AB}}+\overrightarrow{{AC}}+\overrightarrow{{AD}}=4 \overrightarrow{{AO}}+\overrightarrow{{OB}}+\overrightarrow{{OC}}$

Reason $R$ : Polygon law of vector addition yields $\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C D}+\overrightarrow{A D}=2 \overrightarrow{A O}$

In the light of the above statements, choose the most appropriate answer from the options given below

  • [JEE MAIN 2021]

A particle is situated at the origin of a coordinate system. The following forces begin to act on the particle simultaneously (Assuming particle is initially at rest)

${\vec F_1} = 5\hat i - 5\hat j + 5\hat k$            ${\vec F_2} = 2\hat i + 8\hat j + 6\hat k$

${\vec F_3} =  - 6\hat i + 4\hat j - 7\hat k$         ${\vec F_4} =  - \hat i - 3\hat j - 2\hat k$

Then the particle will move

The value of the sum of two vectors $\overrightarrow A $ and $\overrightarrow B $ with $\theta $ as the angle between them is