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Which of the following statements about unit vectors is INCORRECT?
According to the NCERT text, 'A unit vector is a vector of unit magnitude and points in a particular direction. It has no dimension and unit. It is used to specify a direction only.' Therefore, the statement that unit vectors have specific dimensions and units is incorrect.
A vector $\vec{A}$ lies in the x-y plane. If its x-component is $A_x$ and its y-component is $A_y$, how can $\vec{A}$ be expressed in terms of unit vectors $\hat{i}$ and $\hat{j}$?
The NCERT text states, 'Thus, $\vec{A} = A_x \hat{i} + A_y \hat{j}$.' This equation correctly represents a vector in terms of its components along the x and y axes using unit vectors.
If a vector $\vec{A}$ has a magnitude $|A|$ and a unit vector along its direction is $\hat{n}$, which of the following expressions is correct?
According to equation 3.10 in the NCERT text, 'In general, a vector A can be written as $\vec{A} = |A| \hat{n}$ where $\hat{n}$ is a unit vector along A.'
Which of the following is NOT a characteristic of a unit vector?
A unit vector is used to specify direction. Multiplying a unit vector by a scalar changes its magnitude, not its direction (unless the scalar is negative, which reverses the direction). The NCERT states: 'If we multiply a unit vector, say $\hat{n}$ by a scalar $\lambda$, the result is a vector $\lambda \vec{n}$.'
A vector $\vec{A}$ is resolved into two component vectors, $\vec{A_1}$ and $\vec{A_2}$. If $\vec{A_1}$ is parallel to $\hat{i}$ and $\vec{A_2}$ is parallel to $\hat{j}$, which condition must be met for this resolution?
The NCERT text explicitly states: 'We draw lines from the head of A perpendicular to the coordinate axes as in Fig. 3.9(b), and get vectors $\vec{A_1}$ and $\vec{A_2}$ such that $\vec{A_1} + \vec{A_2} = \vec{A}$'.
Consider a vector $\vec{A}$ making an angle $\theta$ with the x-axis. If $A_x$ is its x-component and $A_y$ is its y-component, which of the following relations is correct?
From the summary section, point 10 states: 'If vector A makes an angle $\theta$ with the x-axis, then $A_x = A \cos \theta$, $A_y = A \sin \theta$'.
What is the magnitude of a vector $\vec{A} = A_x \hat{i} + A_y \hat{j}$?
According to point 10 in the summary, the magnitude of a vector $\vec{A} = A_x \hat{i} + A_y \hat{j}$ is $|A| = \sqrt{A_x^2 + A_y^2}$.
If $\vec{r} = x \hat{i} + y \hat{j}$ is the position vector of an object in the x-y plane, what do $x$ and $y$ represent?
As stated in section 3.7.1, for a position vector $\vec{r} = x \hat{i} + y \hat{j}$, '$x$ and $y$ are components of $\vec{r}$ along x-, and y- axes or simply they are the coordinates of the object.'
A vector $\vec{A}$ can be resolved into components along two given vectors $\vec{a}$ and $\vec{b}$ lying in the same plane. This representation is given by:
Point 8 in the summary states: 'A vector A can be resolved into component along two given vectors a and b lying in the same plane: $\vec{A} = \lambda \vec{a} + \mu \vec{b}$ where $\lambda$ and $\mu$ are real numbers.'
What is the primary purpose of unit vectors in a rectangular coordinate system?
The NCERT states, 'A unit vector is a vector of unit magnitude and points in a particular direction. It has no dimension and unit. It is used to specify a direction only.' In a rectangular coordinate system, $\hat{i}$, $\hat{j}$, and $\hat{k}$ specifically point along the x-, y-, and z-axes, respectively, to specify direction.
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