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Class 11
Physics
Motion in a Plane
establish the following vector inequalities geomet...

Question:
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?

Answer:

(a) Let two vectors and be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.

Here, we can write:

(b) Let two vectors and be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.

In a triangle, each side is smaller than the sum of the other two sides.

Therefore, in ΔOMN, we have:

(c) Let two vectors and be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.

In a triangle, each side is smaller than the sum of the other two sides. Therefore, in ΔOPS, we have:

OS < OP + PS

(d) Let two vectors and be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.

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Question:
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?

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Question: 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?

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Question:
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?