Read the passage given below and answer the following questions f… - Vidyadip

Question

Read the passage given below and answer the following questions from (i) to (v). Vectors
Vectors are the physical quantities which have both magnitudes and directions and obey the triangle/parallelogram laws of addition and subtraction. It is specified by giving its magnitude by a number and its direction. e.g. Displacement, acceleration, velocity, momentum, force, etc. A vector is represented by a bold face type and also by an arrow placed over a letter, i.e. F, a, b or $\overrightarrow{\text{F}},\overrightarrow{\text{a}},\overrightarrow{\text{b}}.$ The length of the line gives the magnitude and the arrowhead gives the direction. The point P is called head or terminal point and pointO is called tail or initial point of the vector OP.

Amongst the following quantities, which is not a vector quantity?

Force
Acceleration
Temperature
Velocity

Set of vectors A and B, P and Q are as shown below

Length of A and B is equal, similarly length of P and Q is equal. Then, the vectors which are equal, are:

A and P
P and Q
A and B
B and Q

$\mid\lambda\text{A}\mid=\lambda\mid\text{A}\mid,$ if:

$\lambda>0$
$\lambda,<0$
$\lambda,=0$
$\lambda,\neq0$

Among the following properties regarding null vector which is incorrect?

A + 0 = A
$\lambda0=\lambda$
0A = 0
A - A = 0

The x and y components of a position vector P have numerical values 5 and 6, respectively. Direction and magnitude of vector P are:

$\tan^{-1}\big(\frac{6}{5}\big)\text{and}\sqrt{61}$
$\tan^{-1}\big(\frac{5}{6}\big)\text{and}\sqrt{61}$
60° and 8
30° and 9

✓

Answer

(c) Temperature

Explanation:
Temperature is not a vector quantity because it has magnitude only.
However, force, acceleration and velocity have both a magnitude and a direction.
So, these are vectors in nature.

(c) A and B

Explanation:
Two vectors are said to be equal, if and only if they have the same magnitude and direction.
Among the given vectors A and B are equal vectors as they have same magnitude (length) and direction.
However, P and Q are not equal even though they are of same magnitude because their directions are different.

(c) $\lambda>0$

Explanation:
$\mid\lambda\text{A}\mid=\lambda\mid\text{A}\mid,$ if $\lambda>0$ as multiplication of vector A with a positive number $\lambda$ gives a
vector whose magnitude is changed by the factor $\lambda$ but the direction is same as that of A.

(b) $\lambda0=\lambda$

Explanation:
Null vector 0 is a vector, whose magnitude iszero and its direction cannot be specified.
So, it means, $\mid0\mid=0.$
Thus, $\lambda0=0.$
Hence, property given in option (b) is incorrect.

(c) 60° and 8

Explanation:
Let P be as shown in the figure, then according to the given information

Px = 5, Py = 6
$\therefore\mid\text{P}\mid=\sqrt{\text{P}_\text{x}^2+\text{P}_\text{y}^2}$
$=\sqrt{25+336}$
$\Rightarrow\mid\text{P}\mid=\sqrt{61}$ and $\tan\theta=\frac{\text{P}_\text{y}}{\text{P}_\text{x}}=\frac{6}{5}$
$\Rightarrow\theta=\tan^{-1}\big(\frac{6}{5}\big)$

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