r → = s i ^ + t j ^ is the equation of

a straight line joining the points i ^ and j ^

The position vector of the point of intersection of three planes r → · n → 1 = q 1 , r → · n → 2 = q 2 , r → · n → 3 = q 3 where n → 1 , n → 2 and n → 3 are non-coplanar vectors, is

r → = 1 n → 1 n → 2 n → 3 q 3 n → 1 × n → 2 + q 1 n → 2 × n → 3 + q 2 n → 3 × n → 1

r → = 1 n → 1 n → 2 n → 3 q 1 n → 1 × n → 2 + q 2 n → 2 × n → 3 + q 3 n → 3 × n → 1

r → = 1 n → 3 n → 1 n → 2 q 1 n → 1 × n → 2 + q 2 n → 2 × n → 3 + q 3 n → 3 × n → 1

Find the direction cosines of vector 6 i ^ - 5 j ^ + 7 k ^ .

Given vector is 6 i ^ - 5 j ^ + 7 k ^ . Direction cosines of given vector are 6 ( 6 ) 2 + ( - 5 ) 2 + ( 7 ) 2 , - 5 ( 6 ) 2 + ( - 5 ) 2 + ( 7 ) 2 , 7 ( 6 ) 2 + ( - 5 ) 2 + ( 7 ) 2 i.e., 6 110 , - 5 110 , 7 110

Find the direction cosines of vector 4 i ^ - 5 j ^ + 3 k ^ .

Given vector is 4 i ^ - 5 j ^ + 3 k ^ . Direction cosines of given vector are 4 ( 4 ) 2 + ( - 5 ) 2 + ( 3 ) 2 , - 5 ( 4 ) 2 + ( - 5 ) 2 + ( 3 ) 2 , 3 ( 4 ) 2 + ( - 5 ) 2 + ( 3 ) 2 i.e., 4 5 2 , - 5 5 2 , 3 5 2

Find the direction cosines of vector 5 i ^ - 3 j ^ + 4 k ^ .

Given vector is 5 i ^ - 3 j ^ + 4 k ^ . Direction cosines of given vector are 5 ( 5 ) 2 + ( - 3 ) 2 + ( 4 ) 2 , - 3 ( 5 ) 2 + ( - 3 ) 2 + ( 4 ) 2 , 4 ( 5 ) 2 + ( - 3 ) 2 + ( 4 ) 2 i.e., 5 5 2 , - 3 5 2 , 4 5 2

EASY

Earn 100

What is resolution of a vector?

Important Questions on Forces: Vectors and Moments

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

The angle made by

\vec{r} = 3 \hat{i} + 3 \hat{j}

with the

x

axis is

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

Let a vector

α \hat{i} + β \hat{j}

be obtained by rotating the vector

\sqrt{3} \hat{i} + \hat{j}

by an angle

45 °

about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices

(α, β), (0, β)

and

(0, 0)

is equal to

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

\vec{A}

and

\vec{B}

are two non-zero vectors inclined at an angle

θ

\hat{a}

and

\hat{b}

are unit vectors along

\vec{A}

and

\vec{B}

respectively. The component of

\vec{A}

in the direction of

\vec{B}

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

If y-component of a force acting in x-y plane is $2 \sqrt{3} N$ . Then the x- component will be

Question Image

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

One of the rectangular components of a force of

40 N

20 \sqrt{3} N

. What is the other rectangular component?

MEDIUM

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

The resultant of these forces $\vec{O P}, \vec{O Q}, \vec{O R}, \vec{O S}$ and $\vec{O T}$ is approximately ______ $N .$
[Take $\sqrt{3} = 1.7, \sqrt{2} = 1.4$ Given $\hat{i}$ and $\hat{j}$ unit vectors along $x, y$ axis]

Question Image

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

\vec{r} = s \hat{i} + t \hat{j}

is the equation of

HARD

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

Let

\vec{u}

be a vector on rectangular coordinate system with sloping angle

60 °

. Suppose that,

|\vec{u} - \hat{i}|

is geometric mean of

|\vec{u}|

and

|\vec{u} - 2 \hat{i}|

where

\hat{i}

is the unit vector along

x

-axis then

|\vec{u}|

has the value equal to

\sqrt{a} - \sqrt{b}

where

a, b \in N .

Then the value of

{(a + b)}^{3} + {(a - b)}^{3}

MEDIUM

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

The position vector of the point of intersection of three planes

\vec{r} \cdot {\vec{n}}_{1} = q_{1}, \vec{r} \cdot {\vec{n}}_{2} = q_{2}, \vec{r} \cdot {\vec{n}}_{3} = q_{3}

where

{\vec{n}}_{1}, {\vec{n}}_{2}

and

{\vec{n}}_{3}

are non-coplanar vectors, is

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

Find the direction cosines of vector

6 \hat{i} - 5 \hat{j} + 7 \hat{k}

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

Find the direction cosines of vector

4 \hat{i} - 5 \hat{j} + 3 \hat{k}

MEDIUM

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

A bird flies at an angle of

60^{o}

to the horizontal. Its horizontal component of velocity is

10 {m s}^{- 1} .

Find the vertical component of velocity in

{m s}^{- 1} .

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

The magnitude of resultant of three vectors of magnitude 1, 2 and 3 whose directions are those of the sides of an equilateral triangle taken in order is

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

Magnitude of

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

\sqrt{x}

, write value of

x

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

Given three vectors

\vec{a} = 6 \hat{i} - 3 \hat{j}

\vec{b} = 2 \hat{i} - 6 \hat{j}

and

\vec{c} = - 2 \hat{i} + 21 \hat{j}

such that

\vec{α} = \vec{a} + \vec{b} + \vec{c}

. Then, the resolution of the vector

\vec{α}

into components with respect to

\vec{a}

and

\vec{b}

is given by

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

Find the direction cosines of vector

5 \hat{i} - 3 \hat{j} + 4 \hat{k}

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

If projection of any line on coordinate axes

3, 4

and

5

, then its length is

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

In the figure which of the ways indicated for combining the $x$ and $y$ components of the vector $\vec{a}$ are proper to determine that vector?

Question Image

HARD

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

Given three vectors

\vec{a} = 6 \hat{i} - 3 \hat{j}

\vec{b} = 2 \hat{i} - 6 \hat{j}

and

\vec{c} = - 2 \hat{i} + 21 \hat{j}

such that

\vec{α} = \vec{a} + \vec{b} + \vec{c}

. Then the resolution of the vector

\vec{α}

into components with respect to

\vec{a}

and

\vec{b}

is given by :

EASY

Physics>Mechanics>Forces: Vectors and Moments>Components of Vectors

\vec{a} = 4 \hat{i} + 6 \hat{j}

and

\vec{b} = 3 \hat{j} + 4 \hat{k},

, then the vector form of component of

\vec{a}

along

\vec{b}

What is resolution of a vector?

Important Questions on Forces: Vectors and Moments

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