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

Here, we can write:

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

Therefore, in ΔOMN, we have:

ON < (OM + MN)

If the two vectors and act along a straight line in the same direction, then we can write:

Combining equations (iv) and (v), we get:

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

Here, we have:

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

Therefore, in ΔOMN, we have:

… (iv)

If the two vectors and act along a straight line in the opposite direction, then we can write:

… (v)

Combining equations (iv) and (v), we get:

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

Here we have:

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

If the two vectors act in a straight line but in opposite directions, then we can write:

… (iv)

Combining equations (iii) and (iv), we get:

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

The following relations can be written for the given parallelogram.

The quantity on the LHS is always positive and that on the RHS can be positive or negative. To make both quantities positive, we take modulus on both sides as:

If the two vectors act in a straight line but in the same directions, then we can write:

Combining equations (iv) and (v), we get: