Question

༒ Bʀᴀɪɴʟʏ Sᴛᴀʀs

༒Mᴏᴅᴇʀᴀᴛᴇʀs

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if a = 5i - j -3k and b= i+3j-5k, then show that the vectors a+b and a-b are perpendicular

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Answer 1

Answer:

a+b= 6i+ 2j-8k

a-b= 4i-4j+2k

(a+b).(a-b)=(6i+ 2j-8k).(4i-4j+2k)

= 24-8-16= 24-24=0 =cos®° =cos (90)

so this is proved

Answer 2

Given :

a = 5î - ĵ - 3k

b = î + 3ĵ - 5k

To Show :

(a + b) and (a - b) are perpendicular

Solution:

Firstly let us find the values of (a + b) and (a - b) individually.

For (a + b)

(a + b) = (5î - ĵ - 3k) + (î + 3ĵ - 5k)

(a + b) = (5 + 1)î + (-1 + 3)ĵ + (-3 -5)k

(a + b) = 6î + 2ĵ - 8k

For (a - b)

(a - b) = (5î - ĵ - 3k) - (î + 3ĵ - 5k)

(a - b) = (5 - 1)î + (-1 -3)ĵ + (-3 + 5)k

(a - b) = 4î - 4ĵ + 2k

Now, We know:

If the scalar product of (a + b) and (a - b) comes to be zero, it means both are Perpendicular to each other.

Therefore:

Scalar product = (a + b) . (a - b)

(a + b) . (a - b) = (6î + 2ĵ - 8k) . (4î - 4ĵ + 2k)

(a + b) . (a - b) = (6 × 4) + {2 × (-4)} + {(-8) × 2}

(a + b) . (a - b) = 24 - 8 - 16

(a + b) . (a - b) = 24 - 24

(a + b) . (a - b) = 0

Here, (a + b) . (a - b) = 0 and we know if scalar product of two vectors is 0 then the two vectors are Perpendicular to each other.

Hence Proved that (a + b) and (a - b) are perpendicular.

Hope this is helpful for you ☘

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