Motion in a Lane MCQs for NEET — Physics Questions with Answers

By definition, a unit vector always has a magnitude of 1, regardless of the vector it's associated with. So, the unit vector in the direction of $\vec{A}$ will have a magnitude of 1. While the magnitude of $\vec{A}$ is $\sqrt{3^2 + 4^2} = 5$, the unit vector itself has magnitude 1.

The NCERT text clearly states, 'The quantities $A_x$ and $A_y$ are called x-, and y- components of the vector $\vec{A}$'.

The NCERT text explains: 'We found that the directional aspect of these quantities can be taken care of by + and – signs, as in one dimension only two directions are possible. But in order to describe motion of an object in two dimensions (a plane) or three dimensions (space), we need to use vectors to describe the above-mentioned physical quantities.' This highlights that in more than one dimension, there are more than two possible directions, requiring a more sophisticated tool like vectors.

Following Example 3.4 in the NCERT text:
First, find the velocity vector $\vec{v}(t) = \frac{d\vec{r}}{dt}$:
$\vec{v}(t) = \frac{d}{dt} (3.0t \hat{i} + 2.0t^2 \hat{j} + 5.0 \hat{k}) = 3.0 \hat{i} + 4.0t \hat{j}$
Next, find the acceleration vector $\vec{a}(t) = \frac{d\vec{v}}{dt}$:
$\vec{a}(t) = \frac{d}{dt} (3.0 \hat{i} + 4.0t \hat{j}) = 0 \hat{i} + 4.0 \hat{j} = 4.0 \hat{j}$.

According to equation (3.25) and (3.26) in the NCERT text, '$\Delta\vec{r} = \vec{r}' - \vec{r} = (x'\hat{i} + y'\hat{j}) - (x\hat{i} + y\hat{j}) = (x'-x)\hat{i} + (y'-y)\hat{j}$'.

Summary point 5 states: 'Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$. It also obeys the associative law: $(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$'.

According to the NCERT text section 3.2, 'A scalar quantity is a quantity with magnitude only... Examples are: the distance between two points, mass of an object, the temperature of a body and the time at which a certain event happened.' Velocity, displacement, and acceleration are all vector quantities.

Summary point 11 in the NCERT states: 'Vectors can be conveniently added using analytical method. If sum of two vectors $\vec{A}$ and $\vec{B}$, that lie in x-y plane, is $\vec{R}$, then: $\vec{R} = R_x \hat{i} + R_y \hat{j}$, where, $R_x = A_x + B_x$, and $R_y = A_y + B_y$'.

Equation (3.10) in the NCERT text states: 'In general, a vector $\vec{A}$ can be written as $\vec{A} = |A|\hat{n}$ where $\hat{n}$ is a unit vector along A.' This directly defines $\hat{n}$ as the unit vector along the direction of A.

The NCERT chapter mentions that 'The vector product, however, is not commutative.' While not explicitly stating $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$ in the provided excerpt, this is a fundamental property of the vector (cross) product, which is consistently taught in conjunction with the fact that it's non-commutative. The cross product of two vectors results in a vector perpendicular to the plane containing the two vectors, and reversing the order reverses the direction of the resultant vector (e.g., by the right-hand rule).