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Question
Establish the following vector inequalities geometrically or otherwise:
- | a+b | ≤ |a| + |b|
- |a + b| ≥ | |a| - |b| |
- |a - b| ≤ |a| + |b|
- |a-b| ≥ | |a| - |b| |
When does the equality sign above apply?
Solution
(a) Let two vectors, `veca` and `vecb`, be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.
Here, we can write:
`|vec(OM)| = |veca|` ... (i)
`|vec(MN)| = |vecOP| = |vecb|` ....(ii)
`|vec(ON)| = |veca+ vecb|` .....(iii)
In a triangle, each side is smaller than the sum of the other two sides.
Therefore, in ΔOMN, we have:
ON < (OM + MN)
`|veca + vecb| < |veca|+|vecb|` ....(iv)
If the two vectors `veca` and `vecb` act along a straight line in the same direction, then we can write:
`|veca+vecb| = |veca| + |vecb|` ....(v)
Combining equations (iv) and (v), we get:
`|veca+vecb| <= |veca| + |vecb|`
(b) Let two vectors `veca` and `vecb` be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.
Here, we have:
`|vec(OM)|= |veca|` ...(i)
`|vec(MN)| = |vec(OP)| = |vecb|` ...(ii)
`|vec(ON)| = |veca + vecb|` ...(iii)
In a triangle, each side is smaller than the sum of the other two sides.
Therefore, in ΔOMN, we have:
ON + MN > OM
ON + OM > MN
`|vec(ON)| > |vec(OM) - vec(OP)|` (∵ OP =MN)
`|veca + vecb| > ||veca|-|vecb||` ... (iv)
If the two vectors `veca` and `vecb` act along a straight line in the opposite direction, then we can write:
`|veca+vecb| =||veca|-|vecb||` ... (v)
Combining equations (iv) and (v), we get:
`|veca + vecb| >= ||veca|-|vecb| |`
(c) To prove `|veca - vecb| <= |veca| + |vecb|`
In figure `vec(OL)` and `vec(OM)` represent vectors `veca` and `vecb`, respectively. Here `vec(ON)` represents `veca - vecb`
Consider the Δ OMN
ON < MN + OM
or `|veca-vecb| < |veca| +|-vecb|`
or `|veca - vecb| < |veca| + |vecB|` ...(i)
When `veca` and `vecb` are along the same straight line but point in the opposite direction then
`|veca -vecb| = |veca| + |vecB|` ... (ii)
Combining equations i and ii we get
`|veca-vecb <= |veca| + vec|b|`
(d) To prove `|veca - vecb| >= ||veca| - |vecb||`
Let us consider the ΔOMN
ON + OM > MN or ON > |MN - OM|
Since MN = OL
∴ ON > |OL - OM|
or `|veca - vecb| > ||veca|-|vecb||` ...(i)
When `veca` and `vecb` are along the same straight line and point in the same direction then
`|veca - vecb| = |veca| - |vecb|` ...(ii)
Combining equations i and ii we get
`|veca - vecb| >= ||veca| - |vecb||`
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