Q#1
A situation may be described by using different sets of co-ordinate axes having different orientations. Which of the following do not depend on the orientation of the axes?(a) the value of a scalar
(b) component of a vector
(c) a vector
(d) the magnitude of a vector
Answer: (a), (c), (d)
Sets of co-ordinate axes are simply lines of references to describe the position and orientation of vectors or similar things. Their orientations cannot change (a) the value of scalar, (b) a vector or (d) the magnitude of a vector. But when a vector is resolved along axes, the component is dependent on the angle between the vector and the axis along which it is being resolved. This angle will vary if the orientation of axes is changed. So the component of a vector will depend upon the orientation of the axes.
Q#2
Let vectors C = A + B .
(a) |C| is always greater than |A|
(b) It is always possible to have |C| < |A| and |C| < |B|
(c) C is always equal to A+B
(d) C is never equal to A+B
Answer: (b)
Two vectors are added by "parallelogram law of addition" not by mathematical addition. Vectors A and B are placed with their tails at a point without changing their directions. Taking them as two adjacent sides of a parallelogram, the parallelogram and the diagonal from tails is drawn.
This diagonal gives the sum of the two vectors. Depending on the directions and magnitudes of two vectors, the length of this diagonal (which represents the magnitude of the sum) may be
(i) greater or less than the magnitude of vector A which makes option (a) false,
(ii) equal to or less than the sum of magnitudes of A and B makes option (c) false,
(iii) equal to the sum of magnitudes of A and B if A and B are collinear and having the same direction. That makes the option (d) false.
Only option (b) is true because it is always possible to have |C| < |A| and |C| < |B| . See this condition in the figure below:
Q#3
Let the angle between two nonzero vectors A and B be 120° and its resultant be C.
(a) C must be equal to |A-B|
(b) C must be less than |A-B|
(c) C must be greater than |A-B|
(d) C may be equal to |A-B|
Answer: (c)
|A – B| means the numerical difference of the magnitudes of A and B. The answer may be explained through the following diagram:
Q#3
Let the angle between two nonzero vectors A and B be 120° and its resultant be C.
(a) C must be equal to |A-B|
(b) C must be less than |A-B|
(c) C must be greater than |A-B|
(d) C may be equal to |A-B|
Answer: (c)
|A – B| means the numerical difference of the magnitudes of A and B. The answer may be explained through the following diagram:
In left figure: In triangle OCD angle ODC =120° so angle COD<120°. Hence OC>CD because in a triangle side opposite the greater angle is greater. So C > |A-B|
In right figure: In triangle OCD angle ODC =120° so angle OCD<120°. Hence OC>OD because in a triangle side opposite the greater angle is greater. So C > |A-B|
Q#4
The x-component of the resultant of several vectors
(a) is equal to the sum of the x-components of the vectors
(b) may be smaller than the sum of the magnitudes of the vectors
(c) may be greater than the sum of the magnitudes of the vectors
(d) may be equal to the sum of the magnitudes of the vectors
Answer: (a), (b), (d)
Only (c) is wrong because the sum of the magnitudes of the vectors is the numerical sum while x-component of the resultant vector is the algebraical sum of the x-component of the vectors and will never be greater than the former.
Q#5
The magnitude of the vector product of two vectors A and B may be
(a) greater than AB (b) equal to AB
(c) less than AB (d) equal to zero
Answer: (b), (c), (d)
Magnitude of the cross product of two vectors A and B is ABsinθ and value of sinθ varies between 1 to -1. So ABsinθ can vary between AB to -AB ie it can be either equal to or less than AB or even equal to zero. But it can not be greater than AB.
In right figure: In triangle OCD angle ODC =120° so angle OCD<120°. Hence OC>OD because in a triangle side opposite the greater angle is greater. So C > |A-B|
Q#4
The x-component of the resultant of several vectors
(a) is equal to the sum of the x-components of the vectors
(b) may be smaller than the sum of the magnitudes of the vectors
(c) may be greater than the sum of the magnitudes of the vectors
(d) may be equal to the sum of the magnitudes of the vectors
Answer: (a), (b), (d)
Only (c) is wrong because the sum of the magnitudes of the vectors is the numerical sum while x-component of the resultant vector is the algebraical sum of the x-component of the vectors and will never be greater than the former.
Q#5
The magnitude of the vector product of two vectors A and B may be
(a) greater than AB (b) equal to AB
(c) less than AB (d) equal to zero
Answer: (b), (c), (d)
Magnitude of the cross product of two vectors A and B is ABsinθ and value of sinθ varies between 1 to -1. So ABsinθ can vary between AB to -AB ie it can be either equal to or less than AB or even equal to zero. But it can not be greater than AB.
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