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The angle between vector (\( \vec{A} \)) and (\( \vec{A} - \vec{B} \)) is:

If \(\vec{A}\) -and \(\vec{B}\) -are two vectors satisfying the relation \(\vec{A}\cdot\vec{B} = \left|\vec{A}\times\vec{B}\right|\). Then the value of \(\left|\vec{A}-\vec{B}\right|\) will be:

What will be the projection of vector \( \vec{A} = \hat{i} + \hat{j} + \hat{k} \) on vector \( \vec{B} = \hat{i} + \hat{j} \)?

The resultant of these force $\overrightarrow{OP},\overrightarrow{OQ},\overrightarrow{OR},\overrightarrow{OS} \text{ and } \overrightarrow{OT}$ is approximately_______ N.
$\text{[Take } \sqrt{3}=1.7,\sqrt{2}=1.4 \text{ Given }\hat{i} \text{ and } \hat{j} \text{ unit vectors along } x, y \text{ axis]}$

Two vectors \( \vec{P} \) and \( \vec{Q} \) have equal magnitudes. If the magnitude of \( \vec{P} + \vec{Q} \) is \( n \) times the magnitude of \( \vec{P} - \vec{Q} \), then angle between \( \vec{P} \) and \( \vec{Q} \) is:

A boy reaches the airport and finds that the escalator is not working. He walks up the stationary escalator in time \( t_{1} \). If he remains stationary on a moving escalator then the escalator takes him up in time \( t_{2} \). The time taken by him to walk up on the moving escalator will be:

The magnitude of vectors \( \overrightarrow{OA} \), \( \overrightarrow{OB} \) and \( \overrightarrow{OC} \) in the given figure are equal. The direction of \( \overrightarrow{OA} + \overrightarrow{OB} - \overrightarrow{OC} \) with x-axis will be:

Assertion A : If A, B, C and 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 \( \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} = \overrightarrow{AD} = 2\overrightarrow{AO} \).

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

A person is swimming with a speed of \( 10 \, \text{m/s} \) at an angle of \( 120^\circ \) with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is \(' x' \, \text{m/s} \). The value of \( 'x '\) to the nearest integer is __________ .

A swimmer can swim with velocity of \( 12 \ \text{km/h} \) in still water. Water flowing in a river has velocity \( 6 \ \text{km/h} \). The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his starting point is __________.\(^\circ\) (Round off to the Nearest Integer) (Find the angle in degrees)

Statement I:
If three forces \( \vec{F_1}, \vec{F_2} \) and \( \vec{F_3} \) are represented by three sides of a triangle and \( \vec{F_1} + \vec{F_2} = -\vec{F_3} \), then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement II:
A triangle made up of three forces \( \vec{F_1}, \vec{F_2} \) and \( \vec{F_3} \) as its sides taken in the same order, satisfy the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below:

Three particle \( P, Q \) and \( R \) are moving along the vectors \( \vec{A} = \hat{i} + \hat{j} \), \( \vec{B} = \hat{j} + \hat{k} \) and \( \vec{C} = -\hat{i} + \hat{j} \) respectively. They strike on a point and start to move in different directions. Now particle \( P \) is moving normal to the plane which contains vector \( \vec{A} \) and \( \vec{B} \). Similarly particle \( Q \) is moving normal to the plane which contains vector \( \vec{A} \) and \( \vec{C} \). The angle between the direction of motion of \( P \) and \( Q \) is \( \cos^{-1}\!\left( \dfrac{1}{\sqrt{x}} \right) \). Then the value of \( x \) is _______.

Statement I: Two forces $(\vec{P} + \vec{Q})$ and $(\vec{P} - \vec{Q})$ where $\vec{P} \perp \vec{Q}$, when act at an angle $\theta_1$ to each other, the magnitude of their resultant is $\sqrt{3 \left(P^2 + Q^2\right)}$, when they act at an angle $\theta_2$, the magnitude of their resultant becomes $\sqrt{2 \left(P^2 + Q^2\right)}$. This possible only when $\theta_1 < \theta_2$.
Statement II: In the situation given above $\theta_1 = 60^\circ \text{ and } \theta_2 = 90^\circ$.
In the light of the above statements, choose the most appropriate answer from the options given below:

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JEE Main 2021 26th August Evening Shift

The angle between vector (\( \vec{A} \)) and (\( \vec{A} - \vec{B} \)) is: