Advertisements
Advertisements
प्रश्न
Let \[\vec{a} = 4 \vec{i} + 3 \vec{j} \text { and } \vec{b} = 3 \vec{i} + 4 \vec{j}\]. Find the magnitudes of (a) \[\vec{a}\] , (b) \[\vec{b}\] ,(c) \[\vec{a} + \vec{b} \text { and }\] (d) \[\vec{a} - \vec{b}\].
Advertisements
उत्तर
Given: \[\vec{a} = 4 \vec{i} + 3 \vec{j} \text { and } \vec{b} = 3 \vec{i} + 4 \vec{j}\]
(a) Magnitude of \[\vec{a}\] is given by \[\left| \vec{a} \right| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5\]
Magnitude of \[\vec{b}\] is given by \[\left| \vec{b} \right| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5\]
(c) \[\vec{a} + \vec{b} = (4 \hat {i} + 3 \hat {j} ) + (3 \hat { i} + 4 \hat { j} ) = (7 \hat { i} + 7 \hat {j} )\]
∴ Magnitude of vector \[\vec{a} + \vec{b}\] is given by \[\left| \vec{a} + \vec{b} \right| = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}\]
(d) \[\vec{a} - \vec{b} = \left( 4 \vec{i} + 3 \vec{j} \right) - \left( 3 \vec{i} + 4 \vec{j} \right) = \vec{i} - \vec{j}\]
∴ Magnitude of vector \[\vec{a} - \vec{b}\] is given by \[\left| \vec{a} - \vec{b} \right| = \sqrt{\left( 1 \right)^2 + \left( - 1 \right)^2} = \sqrt{2}\]
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
“Politics is the art of the possible”. Similarly, “Science is the art of the soluble”. Explain this beautiful aphorism on the nature and practice of science.
What are the dimensions of the ratio of the volume of a cube of edge a to the volume of a sphere of radius a?
Suppose a quantity x can be dimensionally represented in terms of M, L and T, that is, `[ x ] = M^a L^b T^c`. The quantity mass
Choose the correct statements(s):
(a) All quantities may be represented dimensionally in terms of the base quantities.
(b) A base quantity cannot be represented dimensionally in terms of the rest of the base quantities.
(c) The dimensions of a base quantity in other base quantities is always zero.
(d) The dimension of a derived quantity is never zero in any base quantity.
Let I = current through a conductor, R = its resistance and V = potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for R and V are \[{\text{ML}}^2 \text{I}^{- 2} \text{T}^{- 3}\] and \[{\text{ML}}^2 \text{T}^{- 3} \text{I}^{- 1}\] respectively.
Test if the following equation is dimensionally correct:
\[V = \frac{\pi P r^4 t}{8 \eta l}\]
where v = frequency, P = pressure, η = coefficient of viscosity.
Test if the following equation is dimensionally correct:
\[v = \frac{1}{2 \pi}\sqrt{\frac{mgl}{I}};\]
where h = height, S = surface tension, \[\rho\] = density, P = pressure, V = volume, \[\eta =\] coefficient of viscosity, v = frequency and I = moment of interia.
Is a vector necessarily changed if it is rotated through an angle?
Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes?
Can a vector have zero component along a line and still have nonzero magnitude?
Is the vector sum of the unit vectors \[\vec{i}\] and \[\vec{i}\] a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?
Let \[\vec{A} = 5 \vec{i} - 4 \vec{j} \text { and } \vec{B} = - 7 \cdot 5 \vec{i} + 6 \vec{j}\]. Do we have \[\vec{B} = k \vec{A}\] ? Can we say \[\frac{\vec{B}}{\vec{A}}\] = k ?
A vector \[\vec{A}\] points vertically upward and \[\vec{B}\] points towards the north. The vector product \[\vec{A} \times \vec{B}\] is
A situation may be described by using different sets coordinate axes having different orientation. Which the following do not depended on the orientation of the axis?
(a) the value of a scalar
(b) component of a vector
(c) a vector
(d) the magnitude of a vector.
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.
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?
A curve is represented by y = sin x. If x is changed from \[\frac{\pi}{3}\text{ to }\frac{\pi}{3} + \frac{\pi}{100}\] , find approximately the change in y.
The electric current in a charging R−C circuit is given by i = i0 e−t/RC where i0, R and C are constant parameters of the circuit and t is time. Find the rate of change of current at (a) t = 0, (b) t = RC, (c) t = 10 RC.
If π = 3.14, then the value of π2 is ______
