You visited us 1 times! Enjoying our articles? Unlock Full Access!

Comparing Vectors

Forces 3 O A ...

Question

Forces 3 O $\vec{A}$ , 5 O $\vec{B}$ act along OA and OB. If their resultant passes through C on AB, then

(a) C is a mid-point of AB
(b) C divides AB in the ratio 2 : 1
(c) 3 AC = 5 CB
(d) 2 AC = 3 CB

Open in App

Solution

(c) 3 AC = 5 CB

Draw ON, the perpendicular to the line AB

Let $\vec{i}$ be the unit vector along ON
The resultant force $\vec{R} = 3 \vec{O A} + 5 \vec{O B} . . . . . (1)$

The angles between $\vec{i}$ and the forces $\vec{R}, 3 \vec{O A}, 5 \vec{O B}$ are ∠CON, ∠AON, ∠BON respectively.

$\vec{R} \cdot \vec{i} = 3 \vec{O A} \cdot \vec{i} + 5 \vec{O B} \cdot \vec{i}$

⇒ R⋅1⋅ cos ∠CON = 3 $\vec{O A}$ ⋅1⋅cos∠AON + 5 $\vec{O B}$ ⋅1⋅cos∠BON

$R \cdot \frac{O N}{O C} = 3 O A \times \frac{O N}{O A} + 5 O B \frac{O N}{O B} \frac{R}{O C} = (3 + 5)$
R = 8 $\vec{O C}$

We know that,

$\vec{O A} = \vec{O C} + \vec{C A} \Rightarrow 3 \vec{O A} = 3 \vec{O C} + 3 \vec{C A} . . . . . (i) \vec{O B} = \vec{O C} + \vec{C B} \Rightarrow 5 \vec{O B} = 5 \vec{O C} + 5 \vec{C B} . . . . . (ii)$

on adding (i) and (ii) we get,

$\begin{matrix} \begin{matrix} 3 \vec{O A} & + & 5 \vec{O B} \end{matrix} & = & 8 \vec{O C} & + & 3 \vec{CA} & + & 5 \vec{CB} \\ \vec{R} & = & 8 \vec{O C} & + & 3 \vec{CA} & + & 5 \vec{CB} \\ 8 \vec{O C} & = & 8 \vec{O C} & + & 3 \vec{CA} & + & 5 \vec{CB} \end{matrix} |3 \vec{AC}| = |5 \vec{CB}| \Rightarrow 3 AC = 5 CB$

Suggest Corrections

Similar questions

Q. Forces

3 \to O A

5 \to O B

act along OA and OB. If their resultant passes through C on AB, then :

Two point charges QA = 2 μ C and QA = -2 μC located 16 cm apart in vacuum. What is the electric field at the mid point O of the line AB joining the two charges?

A, B, C

are fixed points and

P

is a variable point such that the resultant of forces at P represented by

- - \to P A

and

- - \to P B

always passes through

C

D

is the mid-point of

A B

, then locus of

P

is:

Q. Three force

\to P, \to Q

and

\to R

, each of

15

units, act along

A B, B C

and

C A

respectively. The position vectors of

A, B

and

C

are

\to O A = \to 2 i - \to j + \to 3 k

\to O B = \to 5 i + \to 3 j - \to 2 k

and

\to O C = - \to 2 i + \to 2 j + \to 3 k

respectively. The resultant force vector is

Q. O is any point in the interior of ∆ABC. Then, which of the following is true?
(a)

(O A + O B + O C) > (A B + B C + C A)

(b)

(O A + O B + O C) > \frac{1}{2} (A B + B C + C A)

(c)

(O A + O B + O C) < \frac{1}{2} (A B + B C + C A)

(d) None of these

Join BYJU'S Learning Program

Forces 3 O A→, 5 O B→ act along OA and OB. If their resultant passes through C on AB, then (a) C is a mid-point of AB (b) C divides AB in the ratio 2 : 1 (c) 3 AC = 5 CB (d) 2 AC = 3 CB

Forces 3 O $\vec{A}$ , 5 O $\vec{B}$ act along OA and OB. If their resultant passes through C on AB, then

(a) C is a mid-point of AB
(b) C divides AB in the ratio 2 : 1
(c) 3 AC = 5 CB
(d) 2 AC = 3 CB