(i) The magnitude of sum of three vectors A → , B → and C → representing the sides of a cube of length A is equal to (ii) If A → = 3 i ^ + 4 j ^ and B → = 7 i ^ + 24 j ^ then, the vector having the same magnitude as B → and parallel to A → is (iii) If A → ‖ B → then, A → × B → = ? (iv) The magnitude of area of the parallelogram formed by the adjacent sides of vectors A → = 3 i ^ + 2 j ^ and B → = 2 i ^ - 4 k ^ is .. (v) Sum of two opposite vector to each other is a.... vector. (vi) The unit vector along vector i ^ + j ^ + k ^ is ...... (vii) If A → is ...... to B → then, A → · B → = 0 (viii). The vector A → = i ^ + j ^ , where i ^ and j ^ are unit vectors along x -axis and y -axis respectively, makes an angle of ..... degree with x -axis. (ix) If A → + B → + C → = 0 → then, A → · B → × C → = ?

(i) Here A → = B → = B → = A So, the magnitude of the sum of these vectors will be R = A → + B → + C → = A 2 + A 2 + A 2 = 3 A . (ii) Let the required vector be S → whose magnitude is same as B → i.e; S → = B → = 7 2 + 24 2 = 49 + 576 = 625 = 25 , Now, S → = B → . A → A → = 25 × ( 3 i ^ + 4 j ^ ) 3 2 + 4 2 = 25 5 × ( 3 i ^ + 4 j ^ ) = 15 i ^ + 20 j ^ . (iii) As the two vectors are parallel i.e; the angle between them is 0 ∘ , Now, A → × B → = A B sin θ = A B sin 0 ∘ = 0 i.e; a null vector. (iv) Area of the parallelogram will be given as A → × B → , So, A → × B → = i ^ j ^ k ^ 3 2 0 2 0 - 4 = ( 2 × - 4 - 0 ) i ^ - ( 3 × - 4 - 2 × 0 ) j ^ + ( 3 × 0 - 2 × 2 ) k ^ = - 8 i ^ + 12 j ^ - 4 k ^ . Now, A → × B → = ( - 8 ) 2 + 12 2 + ( - 4 ) 2 = 64 + 144 + 16 = 224 units . (v) Negative, Positive or Zero. (vi) Let the given vector be V → = i ^ + j ^ + k , ^ So, the required unit vector will be given as; V ^ = V → V → = i ^ + j ^ + k ^ 1 2 + 1 2 + 1 2 = i ^ + j ^ + k ^ 3 = i ^ 3 + j ^ 3 + k ^ 3 . (vii) A → . B → = A B C o s θ = 0 i.e; the angle between the two vectors is 90 ∘ as C o s 90 ∘ = 0 which means means A → ⊥ B → . (viii) From the plot shown above for the given vector we have; slope = θ = tan - ( 1 1 ) = tan - 1 1 = 45 ° . (ix) As A → + B → + C → = 0 , So, R → = A → + B → + C → = 0 i.e; components of resultant vector are 0 , So A → . ( B → × C → ) = 0 .

E M B I B E

Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-4>Q 13

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Let $\vec{u}$ be a conștant vector and $\vec{v}$ be a vector of constant magnitude such that $|\vec{v}| = \frac{1}{2} |\vec{u}|$ and $|\vec{u}| \neq 0 .$ Then the maximum possible angle between $\vec{u}$ and $\vec{u} + \vec{v}$ is :

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Important Questions on Mathematical Tools in Physics

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Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-1>Q 3

Surface area of sphere as a function of its radius is

A (r) = 4 π r^{2}

the value of

A (10)

will be

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Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-1>Q 5

If

f (x) = x + \frac{1}{x},

then the value of

f (1)

will be

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Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-1>Q 7

If

f (θ) = \sin θ,

find

f (\frac{π}{6})

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Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-1>Q 8

If

f (x) = 5,

then the value of

f (10)

will be

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Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-1>Q 62

Magnitude of

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

is

\sqrt{x}

, write value of

x

.

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Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-1>Q 64

One of the rectangular components of a velocity of $60 km h^{- 1}$ is $30 km h^{- 1}$ . The other rectangular component is $x \sqrt{3} km h^{- 1}$ . Find $x$ .

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Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-1>Q 66

A force of

30 N

is inclined at an angle

θ °

to the horizontal. If its vertical component is

18 N

, and the horizontal component is

x N

. What is the value of

x + 2 θ

?

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Alpha Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>Exercise-4>Q 14

(i) The magnitude of sum of three vectors $\vec{A}, \vec{B}$ and $\vec{C}$ representing the sides of a cube of length A is equal to

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(ii) If $\vec{A} = 3 \hat{i} + 4 \hat{j}$ and $\vec{B} = 7 \hat{i} + 24 \hat{j}$ then, the vector having the same magnitude as $\vec{B}$ and parallel to
$\vec{A}$ is
(iii) If $\vec{A} ‖ \vec{B}$ then, $\vec{A} \times \vec{B} = ?$
(iv) The magnitude of area of the parallelogram formed by the adjacent sides of vectors $\vec{A} = 3 \hat{i} + 2 \hat{j}$ and $\vec{B} = 2 \hat{i} - 4 \hat{k}$ is ..
(v) Sum of two opposite vector to each other is a.... vector.
(vi) The unit vector along vector $\hat{i} + \hat{j} + \hat{k}$ is ......
(vii) If $\vec{A}$ is ...... to $\vec{B}$ then, $\vec{A} \cdot \vec{B} = 0$
(viii). The vector $\vec{A} = \hat{i} + \hat{j},$ where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ -axis and $y$ -axis respectively, makes an angle of ..... degree with $x$ -axis.
(ix) If $\vec{A} + \vec{B} + \vec{C} = \vec{0}$ then, $\vec{A} \cdot (\vec{B} \times \vec{C}) = ?$