Establish the following vector inequalities geometrically or otherwise : (a) |a+b| < |a| + |b| (b) |a+b| > ||a| −|b|| (c) |a−b| < |a| + |b| (d) |a−b| > ||a| − |b|| When does the equality sign above apply?
a)
The figure below shoes the parallelogram, whose adjacent sides are two vectors, say a and b , the diagonal of the parallelogram represents the vector a + b .
From parallelogram OMNP,
OM = PN = | a | OP = MN = | b | ON = | a + b |
From properties of triangle, the length of each side of triangle is less than the sum of other two sides of the triangle.
In Δ OMN ,
ON < ( OM + ON )
Substitute the values in the above expression.
| a + b | < | a | + | b | (I)
If a and b acts along the straight line in same direction, then
| a + b | = | a | + | b | (II)
From equation (I) and equation (II),
| a + b | ≤ | a | + | b |
Thus, it is proved that | a + b | ≤ | a | + | b | .
b)
The figure below shoes the parallelogram, whose adjacent sides are two vectors, say a and b , the diagonal of the parallelogram represents the vector a + b .
From parallelogram OMNP,
OM = PN = | a | OP = MN = | b | ON = | a + b |
From properties of triangle, the length of each side of triangle is less than the sum of other two sides of the triangle.
In Δ OMN ,
ON+OM > MN | ON | > | MN − OM |
Substitute the values in the above expression.
| | a + b | | > | | a | − | b | | | a + b | > | | a | − | b | | (III)
If a and b acts along the straight line in same direction, then
| a + b | = | | a | − | b | | (IV)
From equation (III) and equation (IV),
| a + b | ≥ | | a | − | b | |
Thus, it is proved that | a + b | ≥ | | a | − | b | | .
c)
The figure below shoes the parallelogram, whose adjacent sides are two vectors, say a and − b , the diagonal of the parallelogram represents the vector a − b .
From parallelogram OPSR,
OR = PS = | b | OP = | a | OS = | a − b |
From properties of triangle, the length of each side of triangle is less than the sum of other two sides of the triangle.
In Δ OPS ,
OS < OP + PS
Substitute the values in the above expression.
| a − b | < | a | + | b | (V)
If a and b acts along the straight line in same direction, then
| a + b | = | | a | − | b | | (VI)
From equation (V) and equation (VI),
| a − b | ≤ | a | + | b |
Thus, it is proved that | a − b | ≤ | a | + | b | .
d)
The figure below shoes the parallelogram, whose adjacent sides are two vectors, say a and − b , the diagonal of the parallelogram represents the vector a − b .
From parallelogram OPSR,
OR = PS = | b | OP = | a | OS = | a − b |
From properties of triangle, the length of each side of triangle is less than the sum of other two sides of the triangle.
In Δ OPS ,
OS+PS > OP OS > OP − PS | OS | > | OP − PS |
Substitute the values in the above expression.
| | a − b | | > | | a | − | b | | | a − b | > | | a | − | b | | (VII)
If a and b acts along the straight line in same direction, then
| a − b | = | | a | − | b | | (VIII)
From equation (VII) and equation (VIII),
| a − b | ≥ | | a | − | b | |
Thus, it is proved that | a − b | ≥ | | a | − | b | | .
a)
The figure below shoes the parallelogram, whose adjacent sides are two vectors, say
From parallelogram OMNP,
From properties of triangle, the length of each side of triangle is less than the sum of other two sides of the triangle.
In
Substitute the values in the above expression.
If
From equation (I) and equation (II),
Thus, it is proved that
b)
The figure below shoes the parallelogram, whose adjacent sides are two vectors, say
From parallelogram OMNP,
From properties of triangle, the length of each side of triangle is less than the sum of other two sides of the triangle.
In
Substitute the values in the above expression.
If
From equation (III) and equation (IV),
Thus, it is proved that
c)
The figure below shoes the parallelogram, whose adjacent sides are two vectors, say
From parallelogram OPSR,
From properties of triangle, the length of each side of triangle is less than the sum of other two sides of the triangle.
In
Substitute the values in the above expression.
If
From equation (V) and equation (VI),
Thus, it is proved that
d)
The figure below shoes the parallelogram, whose adjacent sides are two vectors, say
From parallelogram OPSR,
From properties of triangle, the length of each side of triangle is less than the sum of other two sides of the triangle.
In
Substitute the values in the above expression.
If
From equation (VII) and equation (VIII),
Thus, it is proved that
