Establish the following vector inequalities geometrically or otherwise:

(a) $$\left | a + b \right |\leq \left | a \right | + \left | b \right |$$

(b) $$|a-b|\geq||a|-|b||$$

(c) $$\left | a – b \right |\leq \left | a \right | + \left | b \right |$$

(d) $$| a – b |\geq | | a | – | b | |$$

When does the equality sign above apply?

Asked by Pragya Singh | 1 year ago |  109

##### Solution :-

(a) Let two vectors $$\vec{a}\;and \;\vec{b}$$

be represented by the adjacent sides of a parallelogram PQRS,

Here,

$$\left | \vec{QR} \right | = \left | \vec{a} \right |$$........(i)

$$\left | \vec{RS} \right | = \left | \vec{QP} \right | = \left | \vec{b} \right |$$..........(ii)

$$\left | \vec{QS} \right | = \left | \vec{a} + \vec{b} \right |$$ ............(iii)

Each side in a triangle is smaller than the sum of the other two sides.

Therefore, in $$\Delta QRS,$$

QS < (QR + RS)

$$\left | \vec{a} + \vec{b} \right | < \left | \vec{a} \right | + \left | \vec{b} \right |$$ ........(iv)

If the two vectors $$\vec{a}$$ and $$\vec{b}$$act along a straight line in the same direction, then:

$$\left | \vec{a} + \vec{b} \right | = \left | \vec{a} \right | + \left | \vec{b} \right |$$ ...........(v)

Combine equation (iv) and (v),

$$\left | \vec{a} + \vec{b} \right | \leq \left | \vec{a} \right | + \left | \vec{b} \right |$$

(b) Let two vectors $$\vec{a}\;and \;\vec{b}$$ be represented by the adjacent sides of a parallelogram PQRS, as given in the figure.

Here,

$$\left | \vec{QR} \right | = \left | \vec{a} \right |$$ ........(i)

$$\left | \vec{RS} \right | = \left | \vec{QP} \right | = \left | \vec{b} \right |$$ ..........(ii)

$$\left | \vec{QS} \right | = \left | \vec{a} + \vec{b} \right |$$ ..........(iii)

Each side in a triangle is smaller than the sum of the other two sides.

Therefore, in $$\Delta QRS$$,

QS + RS > QR

QS + QR > RS

$$\left | \vec{QS} \right | > \left | \vec{QR} – \vec{QP} \right | (QP = RS)$$

$$| \vec{a} + \vec{b} |> | | \vec{a} | – | \vec{b} | |$$ ...............(iv)

If the two vectors $$\vec{a}$$ and $$\vec{b}$$  act along a straight line in the same direction, then:

$$\vec{a} + \vec{b} | = | | \vec{a} | – | \vec{b} | |$$ ..........(v)

Combine equation (iv) and (v):

$$| \vec{a} + \vec{b} | \geq | | \vec{a} | – | \vec{b} |$$

(c) Let two vectors $$\vec{a}$$ and $$\vec{b}$$ be represented by the adjacent sides of a parallelogram PQRS, as given in the figure.

Here,

$$\left | \vec{PQ} \right | = \left | \vec{SR} \right | = \left | \vec{b} \right |$$—– (i)

$$\left | \vec{PS} \right | = \left | \vec{a} \right |$$—– (ii)

Each side in a triangle is smaller than the sum of the other two sides.

Therefore, in $$\Delta PSR,$$

PR < PS + SR

$$\left | \vec{a} – \vec{b} \right | < \left | \vec{a} \right | + \left |- \vec{b} \right | \left | \vec{a} – \vec{b} \right | < \left | \vec{a} \right | + \left | \vec{b} \right |$$ —– (iii)

If the two vectors act along a straight line in the opposite direction, then:

$$\left | \vec{a} – \vec{b} \right | = \left | \vec{a} \right | + \left | \vec{b} \right |$$ —– (iv)

Combine (iii) and (iv),

$$\left | \vec{a} – \vec{b} \right | \leq \left | \vec{a} \right | + \left | \vec{b} \right |$$

(d) Let two vectors $$\vec{a}\;and \;\vec{b}$$be represented by the adjacent sides of a parallelogram PQRS, as given in the figure.

Here,

PR + SR > PS —– (i)
PR > PS – SR —– (ii)

$$\left | \vec{a} – \vec{b} \right | > \left | \vec{a} \right | – \left | \vec{b} \right |$$ —– (iii)

The quantity on the left hand side is always positive and that on the right hand side can be positive or negative. We take modulus on both the sides to make both quantities positive:

$$\left |\left | \vec{a} – \vec{b} \right | \right | > \left |\left | \vec{a} \right | – \left | \vec{b} \right | \right | \left | \vec{a} – \vec{b} \right | > \left |\left | \vec{a} \right | – \left | \vec{b} \right | \right |$$—–  (iv)

If the two vectors act along a straight line in the opposite direction, then:

$$\left | \vec{a} – \vec{b} \right | = \left |\left | \vec{a} \right | – \left | \vec{b} \right | \right |$$—– (v)

Combine (iv) and (v):

$$\left | \vec{a} – \vec{b} \right | \geq \left |\left | \vec{a} \right | – \left | \vec{b} \right | \right |$$

Answered by Abhisek | 1 year ago

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