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When determining the angle $\theta$ for the vector product $\vec{a} \times \vec{b}$, which range of angles should be considered?
The NCERT text states: 'While applying either of the above rules, the rotation should be taken through the smaller angle ($<180^\circ$) between $\vec{a}$ and $\vec{b}$'.
The vector product is also known as the:
The NCERT text explicitly states: 'Because of the cross ($\times$) used to denote the vector product, it is also referred to as cross product.'
Which of the following statements is TRUE regarding the vector product?
The NCERT text highlights: 'The vector product, however, is not commutative.' It further illustrates this with '$\vec{j} \times \vec{i} = -\vec{k}$' whereas '$\vec{i} \times \vec{j} = \vec{k}$'.
If $\vec{a} = 3\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{b} = -2\hat{j} + 3\hat{k}$, what is the value of $\vec{a} \cdot \vec{b}$?
The scalar product (dot product) is calculated as $A_x B_x + A_y B_y + A_z B_z$. Given $\vec{a} = 3\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{b} = 0\hat{i} - 2\hat{j} + 3\hat{k}$, then $\vec{a} \cdot \vec{b} = (3)(0) + (4)(-2) + (-5)(3) = 0 - 8 - 15 = -23$. This is based on Example 6.4 which asks for both scalar and vector product, but only shows the calculation for scalar product (dot product).
For unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$, which of the following is correct?
The NCERT text lists the results: '(i) $\hat{i} \times \hat{i} = 0, \hat{j} \times \hat{j} = 0, \hat{k} \times \hat{k} = 0$' and '(ii) $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$.' Based on this, option o3 is correct.
If $\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ and $\vec{b} = b_x\hat{i} + b_y\hat{j} + b_z\hat{k}$, the vector product $\vec{a} \times \vec{b}$ can be expressed in determinant form as:
The NCERT text provides the determinant form: '$\vec{a} \times \vec{b} = |\begin{smallmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{smallmatrix}|$.'
The time rate of change of the angular momentum of a particle is equal to:
The NCERT text states: 'Thus, the time rate of change of the angular momentum of a particle is equal to the torque acting on it. This is the rotational analogue of the equation $\vec{F} = d\vec{p}/dt...$'
If $\vec{r}$ is the position vector and $\vec{p}$ is the linear momentum of a particle, its angular momentum $\vec{l}$ is given by:
The NCERT text defines angular momentum: '$\vec{l} = \vec{r} \times \vec{p}$.'
If $\vec{v}$ is the linear velocity and $\vec{\omega}$ is the angular velocity for a particle in rotational motion, their relation is given by:
The NCERT text mentions: 'The linear velocity of the particle at P is $\vec{v} = \vec{\omega} \times \vec{r}$'.
The angular velocity vector $\vec{\omega}$ for rotation about a fixed axis lies:
The NCERT text states: 'For rotation about a fixed axis, the angular velocity vector lies along the axis of rotation, and points out in the direction in which a right handed screw would advance...'
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The bank mixes NEET previous year questions (PYQs) with practice questions, each tagged with its exam appearances where applicable.