Which of the following is not true about vectors A → , B → and C → ?

A → × C → × B → × C → is a scalar value.

A → · A → B → · C → is a scalar value.

A → × B → · B → × C → is a scalar value.

A → × B → × C → is a vector value.

If a → + b → + c → = 0 → , then prove that a → × b → = b → × c → = c → × a → .

Given that, a → + b → + c → = 0 → and To prove: a → × b → = b → × c → = c → × a → Here, a → = - ( b → + c → ) ⇒ a → × b → = - ( b → + c → ) × b → = - b → × b → - c → × b → = 0 → + b → × c → = b → × c → [Using a → × a → = b → × b → = c → × c → = 0 ] Again, b → = - ( a → + c → ) ⇒ a → × b → = - a → × ( a → + c → ) = - a → × a → - a → × c → = 0 - a → × c → = c → × a → Thus, a → × b → = b → × c → = c → × a → Hence, proved.

MEDIUM

Earn 100

Describe moment of couple.

Important Questions on Systems of Particles and Rotational Motion

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{A}

and

\vec{B}

are two non-zero vectors inclined at an angle

θ

\hat{a}

and

\hat{b}

are unit vectors along

\vec{A}

and

\vec{B}

respectively. The component of

\vec{A}

in the direction of

\vec{B}

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

The vector sum of two forces is perpendicular to their vector differences. In that case, the forces

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

Resultant of two vectors

\vec{P}

and

\vec{Q}

is of magnitude

R_{1} .

If direction of

\vec{Q}

is reversed, the resultant is of magnitude

R_{2}

. The value of

(R_{1}^{2} + R_{2}^{2})

[\cos (π - θ) = - \cos θ]

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

In a triangle

A B C

, the sides

A B

and

A C

are represented by the vectors

3 \hat{i} + \hat{j} + \hat{k}

and

\hat{i} + 2 \hat{j} + \hat{k},

respectively. Calculate the angle

∠ A B C .

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{A}

is a vector quantity such that

|\vec{A}| =

non-zero constant. Which of the following expression is true for

\vec{A}

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

The unit vector

(a \hat{i} + b \hat{j})

is perpendicular to

(\hat{i} + \hat{j}) .

The value of

b

can be:

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

Vector

\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k}

and

\vec{b} = \hat{i} - \hat{j} + \hat{k}

. What is the unit vector along

\vec{a} + \vec{b}

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

Consider the vectors

\vec{A} = (\hat{i} + \hat{j} - \hat{k})

\vec{B} = (2 \hat{i} - \hat{j} + \hat{k})

and

\vec{C} = \frac{1}{\sqrt{5}} (\hat{i} - 2 \hat{j} + 2 \hat{k})

. What is the value of

\vec{C} \cdot (\vec{A} \times \vec{B})

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

Which of the following is not true about vectors

\vec{A}, \vec{B}

and

\vec{C}

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{P} = 3 \hat{i} + \sqrt{3} \hat{j} + 2 \hat{k}

and

\vec{Q} = 4 \hat{i} + \sqrt{3} \hat{j} + 2.5 \hat{k}

then, the unit vector in the direction of

\vec{P} \times \vec{Q}

\frac{1}{x} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})

. The value of

x

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

In an octagon $A B C D E F G H$ of equal side, what is the sum of $\vec{A B} + \vec{A C} + \vec{A D} + \vec{A E} + \vec{A F} + \vec{A G} + \vec{A H},$ if, $\vec{AO} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$

Question Image

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{A}

and

\vec{B}

are two vectors satisfying the relation

\vec{A} \cdot \vec{B} = | \vec{A} \times \vec{B} |

. Then the value of

| \vec{A} - \vec{B} |

will be:

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

A triangular plate is shown. A force $\vec{F} = 4 \hat{i} - 3 \hat{j}$ is applied at point $P .$ The torque at point $P$ with respect to point $O$ and $Q$ are:

Question Image

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{A} = 3 \hat{i} - 2 \hat{j} + \hat{k}, \vec{B} = \hat{i} - 3 \hat{ȷ} + 5 \hat{k}

and

\vec{C} = 2 \hat{i} + \hat{j} - 4 \hat{k}

form a right angled triangle then out of the following which one is satisfied?

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

Consider a frame that is made up of two thin massless rods $A B$ and $A C$ as shown in the figure. A vertical force $\vec{P}$ of magnitude $100 N$ is applied at point $A$ of the frame.

Question Image

Suppose the force is $\vec{P}$ resolved parallel to the arms $A B$ and $A C$ of the frame. The magnitude of the resolved component along the arm $A C$ is $x N$ . The value of $x$ , to the nearest integer, is ________.

[Given : $\sin (35 °) = 0.573, \cos (35 °) = 0.819, \sin (110 °) = 0.939, \cos (110 °) = - 0.342$ ]

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{a} + \vec{b} + \vec{c} = \vec{0}

, then prove that

\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

If the angle between the vectors

\vec{P}

and

\vec{Q}

θ,

the value of

(\vec{Q} \times \vec{P}) \cdot \vec{P}

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{P} = 2 \hat{ı} - 2 \hat{j} + 3 \hat{k}

and

\vec{Q} = \hat{ı} + \hat{j} + 2 \hat{k}

, then cosine of angle between

\vec{P}

and

\vec{Q}

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}

, then the angle between

\vec{P}

and

\vec{Q}

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>The Vector Product of Vectors

\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}

, the angle between

\vec{P}

and

\vec{Q}

θ (0 ° < θ < 360 °)

. The value of

θ

will be ___

°

Describe moment of couple.

Important Questions on Systems of Particles and Rotational Motion

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