Advertisements
Advertisements
प्रश्न
Two vectors have magnitudes 2 unit and 4 unit respectively. What should be the angle between them if the magnitude of the resultant is (a) 1 unit, (b) 5 unit and (c) 7 unit.
Advertisements
उत्तर
Let the two vectors be \[\vec{a}\] and \[\vec{b}\]
Now,
\[\left| \vec{a} \right| = 3 \text { and } \left| \vec{b} \right| = 4\]
\[\sqrt{\vec{a}^2 + \vec{b}^2 + 2 . \vec{a} . \vec{b} \cos \theta} = 1\]
\[ \Rightarrow \sqrt{3^2 + 4^2 + 2 . 3 . 4 \cos \theta} = 1\]
Squaring both sides, we get:
\[25 + 24 \cos \theta = 1\]
\[ \Rightarrow 24 \cos \theta = - 24\]
\[ \Rightarrow \cos \theta = - 1\]
\[ \Rightarrow \theta = 180^\circ\]
Hence, the angle between them is 180°.
(b) If the resultant vector is 5 units, then
\[\sqrt{\vec{a}^2 + \vec{b}^2 + 2 . \vec{a} . \vec{b} \cos \theta} = 5\]
\[ \Rightarrow \sqrt{3^2 + 4^2 + 2 . 3 . 4 \cos \theta} = 5\]
Squaring both sides, we get:
25 + 24 cos θ = 25
⇒ 24 cos θ = 0
⇒ cos θ = 90°
Hence, the angle between them is 90°.
(c) If the resultant vector is 7 units, then
\[\sqrt{\vec{a}^2 + \vec{b}^2 + 2 . \vec{a} . \vec{b} \cos \theta} = 1\]
\[ \Rightarrow \sqrt{3^2 + 4^2 + 2 . 3 . 4 \cos \theta} = 7\]
Squaring both sides, we get:
25 + 24 cos θ = 49,
⇒ 24 cos θ = 24
⇒ cos θ = 1
⇒ θ = cos−1 1 = 0°
Hence, the angle between them is 0°.
APPEARS IN
संबंधित प्रश्न
“Every great physical theory starts as a heresy and ends as a dogma”. Give some examples from the history of science of the validity of this incisive remark
“It is more important to have beauty in the equations of physics than to have them agree with experiments”. The great British physicist P. A. M. Dirac held this view. Criticize this statement. Look out for some equations and results in this book which strike you as beautiful.
If all the terms in an equation have same units, is it necessary that they have same dimensions? If all the terms in an equation have same dimensions, is it necessary that they have same units?
Suggest a way to measure the thickness of a sheet of paper.
\[\int\frac{dx}{\sqrt{2ax - x^2}} = a^n \sin^{- 1} \left[ \frac{x}{a} - 1 \right]\]
The value of n is
Find the dimensions of frequency .
Find the dimensions of magnetic permeability \[\mu_0\]
The relevant equation are \[F = qE, F = qvB, \text{ and }B = \frac{\mu_0 I}{2 \pi a};\]
where F is force, q is charge, v is speed, I is current, and a is distance.
Find the dimensions of Planck's constant h from the equation E = hv where E is the energy and v is the frequency.
Test if the following equation is dimensionally correct:
\[v = \sqrt{\frac{P}{\rho}},\]
where v = velocity, ρ = density, P = pressure
Let \[\vec{A} = 3 \vec{i} + 4 \vec{j}\]. Write a vector \[\vec{B}\] such that \[\vec{A} \neq \vec{B}\], but A = B.
If \[\vec{A} \times \vec{B} = 0\] can you say that
(a) \[\vec{A} = \vec{B} ,\]
(b) \[\vec{A} \neq \vec{B}\] ?
The resultant of \[\vec{A} \text { and } \vec{B}\] makes an angle α with \[\vec{A}\] and β with \[\vec{B}\],
A vector \[\vec{A}\] makes an angle of 20° and \[\vec{B}\] makes an angle of 110° with the X-axis. The magnitudes of these vectors are 3 m and 4 m respectively. Find the resultant.
Let \[\vec{A} \text { and } \vec{B}\] be the two vectors of magnitude 10 unit each. If they are inclined to the X-axis at angle 30° and 60° respectively, find the resultant.
A carrom board (4 ft × 4 ft square) has the queen at the centre. The queen, hit by the striker moves to the from edge, rebounds and goes in the hole behind the striking line. Find the magnitude of displacement of the queen (a) from the centre to the front edge, (b) from the front edge to the hole and (c) from the centre to the hole.
Suppose \[\vec{a}\] is a vector of magnitude 4.5 units due north. What is the vector (a) \[3 \vec{a}\], (b) \[- 4 \vec{a}\] ?
If \[\vec{A} = 2 \vec{i} + 3 \vec{j} + 4 \vec{k} \text { and } \vec{B} = 4 \vec{i} + 3 \vec{j} + 2 \vec{k}\] find \[\vec{A} \times \vec{B}\].
If \[\vec{A} , \vec{B} , \vec{C}\] are mutually perpendicular, show that \[\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0\] Is the converse true?
Jupiter is at a distance of 824.7 million km from the Earth. Its angular diameter is measured to be 35.72˝. Calculate the diameter of Jupiter.
