Question

2.
Let a = i + j + pk and b = i +j+k, then |a+b|=| a | + | b |, holds for
(a) All real p
(bh No real p
(c) p =-1
(d) p=1
​

Answer 1

Answer:

No real p all are complex

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Answer 2

|a+b|=| a | + | b | holds for p = 1.

Option(d) is correct.

Given:

a = i + j + pk and b = i +j+k

To Find:

We are required to find the values of p hold for |a+b|=| a | + | b |

Solution:

We are given |a+b|=| a | + | b | ------(1)

a = i + j + pk , b = i + j + k

a+b = 2i + 2j + (p+1)k

|a+b| = √2²+2²+(p+1)²

        = √4+4+p²+2p+1

        = √p²+2p+9

| a | = √1²+1²+p²

| a | = √2+p²

| b | = √1²+1²+1²

| b | = √3

Now substitute the values of | a |, |b|, and |a+b| in the equation(1)

√p²+2p+9 = √2+p² + √3

On squaring both sides

p²+2p+9 = ((√2+p²) + √3)²

p²+2p+9 = 2+p² + 3 + 2(√6+3p²)

2(p+2) = 2(√6+3p²)

p+2 = (√3p²+6)

On squaring both sides

(p+2)² = (√3p²+6)²

p²+4p+4 = 3p²+6

2p²-4p+2 + 0

p²-2p+1 = 0

(p-1)² = 0

p = 1

Therefore, |a+b|=| a | + | b | holds for p = 1.

Hence, Option(d) is correct.

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