Advertisements
Advertisements
प्रश्न
Let the angle between two nonzero vectors \[\vec{A}\] and \[\vec{B}\] be 120° and its resultant be \[\vec{C}\].
पर्याय
C must be equal to \[\left| A - B \right|\]
C must be less than \[\left| A - B \right|\]
C must be greater than \[\left| A - B \right|\]
C may be equal to \[\left| A - B \right|\]
Advertisements
उत्तर
C must be less than \[\left| A - B \right|\]
Here, we have three vector A, B and C.
\[\left| \vec{A} + \vec{B} \right|^2 = \left| \vec{A} \right|^2 + \left| \vec{B} \right|^2 + 2 \vec{A} . \vec{B} . . . (i)\]
\[ \left| \vec{A} - \vec{B} \right|^2 = \left| \vec{A} \right|^2 + \left| \vec{B} \right|^2 - 2 \vec{A} . \vec{B} . . . (ii)\]
Subtracting (i) from (ii), we get:
\[\left| \vec{A} + \vec{B} \right|^2 - \left| \vec{A} - \vec{B} \right|^2 = 4 \vec{A} . \vec{B}\]
Using the resultant property \[\vec{C} = \vec{A} + \vec{B}\],we get:
\[\left| \vec{C} \right|^2 - \left| \vec{A} - \vec{B} \right|^2 = 4 \vec{A} . \vec{B} \]
\[ \Rightarrow \left| \vec{C} \right|^2 = \left| \vec{A} - \vec{B} \right|^2 + 4 \vec{A} . \vec{B} \]
\[ \Rightarrow \left| \vec{C} \right|^2 = \left| \vec{A} - \vec{B} \right|^2 + 4\left| \vec{A} \right|\left| \vec{B} \right|\cos120^\circ\]
Since cosine is negative in the second quadrant, C must be less than \[\left| A - B \right|\] .
APPEARS IN
संबंधित प्रश्न
Some of the most profound statements on the nature of science have come from Albert Einstein, one of the greatest scientists of all time. What do you think did Einstein mean when he said : “The most incomprehensible thing about the world is that it is comprehensible”?
“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
Suggest a way to measure the thickness of a sheet of paper.
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.
Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.
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.
Is a vector necessarily changed if it is rotated through an angle?
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?
A vector is not changed if
The resultant of \[\vec{A} \text { and } \vec{B}\] makes an angle α with \[\vec{A}\] and β with \[\vec{B}\],
A vector \[\vec{A}\] points vertically upward and \[\vec{B}\] points towards the north. The vector product \[\vec{A} \times \vec{B}\] is
Let \[\vec{C} = \vec{A} + \vec{B}\]
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.
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) \[\vec{a} - \vec{b}\].
Let A1 A2 A3 A4 A5 A6 A1 be a regular hexagon. Write the x-components of the vectors represented by the six sides taken in order. Use the fact the resultant of these six vectors is zero, to prove that
cos 0 + cos π/3 + cos 2π/3 + cos 3π/3 + cos 4π/3 + cos 5π/3 = 0.
Use the known cosine values to verify the result.
Write the number of significant digits in (a) 1001, (b) 100.1, (c) 100.10, (d) 0.001001.
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.
