Test Index

Motion in a Plane

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Question : 7
Total: 32
Given
→
a
+
→
b
+
→
c
+
→
d
=0 , which of the following statements are correct
(a)
→
a
,
→
b
,
→
c
 and
→
d
 must each be a null vector,
(b) The magnitude of (
→
a
+
→
c
) equals the magnitude of (
→
b
+
→
d
) ,
(c) The magnitude of
→
a
 can never be greater than the sum of the magnitudes of
→
b
,
→
c
 and
→
d

(d)
→
b
+
→
c
 must lie in the plane of
→
a
 and
→
d
 , if
→
a
 and
→
d
 are not collinear, and in the line of
→
a
 and
→
d
, if they are collinear?
Solution:  
(i) The statement that
→
a
,
→
b
,
→
c
 and
→
d
 must each be a null vector, if
→
a
+
→
b
+
→
c
+
→
d
=0 ; is not correct. It is because
→
a
+
→
b
+
→
c
+
→
d
 can be zero in many ways other than that
→
a
,
→
b
,
→
c
and
→
d
 must each be a null vector.
(ii) Since
→
a
+
→
b
+
→
c
+
→
d
=0 ;
(
→
a
+
→
c
) = − (
→
b
+
→
d
)
Thus, vector (
→
a
+
→
c
) is equal to negative of vector (
→
b
+
→
d
) and hence the statement that magnitudeof (
→
a
+
→
c
) is equal to the magnitude of (
→
b
+
→
d
) is correct.
(iii) since‌
→
a
+
→
b
+
→
c
+
→
d
=0;
→
a
=−(
→
b
+
→
c
+
→
d
)
Therefore, magnitude of vector
→
a
 is equal to magnitudeof vector (
→
b
+
→
c
+
→
d
). The sum of the magnitudes of vectors
→
b
,
→
c
 and
→
d
 may be greater than or equal to that of vector
→
a
. Hence, the statement that the magnitude of
→
a
 can never be greater than the sum of the magnitudes of
→
b
,
→
c
 and
→
d
 is correct.
(iv) ‌since‌
→
a
+
→
b
+
→
c
+
→
d
=0;
(
→
b
+
→
c
)+
→
a
+
→
d
=0
The resultant sum of the three vectors
→
b
+
→
c
,
→
a
 and
→
d
 can be zero only if
→
b
+
→
c
 is in the plane of
→
a
 and
→
d
. In case, the vectors
→
a
 and
→
d
 are collinear,
→
b
+
→
c
 must be in line of
→
a
 and
→
d
Hence, the given statement is correct.
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