Question

If a = i+j and b= 2i-k then the point of intersection of lines r x a = b x a and r x b = a x b is

Answer 1

point of intersection of lines is 3i + j - k

it is given that, a = i + j and b = 2i - k

r × a = b × a

⇒(r - b) × a = 0

so, r = b + λ a

= (2i - k) + λ(i + j)

= (2 + λ)i + λ j - k.......(1)

similarly, r × b = a × b

⇒(r - a) × b = 0

⇒r = a + η b

= (i + j) + η(2i - k)

= (1 + 2η)i + j - η k ......(2)

from equations (1) and (2),

2 + λ = 1 + 2η

λ = 1 and η = 1 putting these in equation (1) or (2)

so, r = 3i + j - k

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Answer 2

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it is given that, a = i + j and b = 2i - k

r × a = b × a

⇒(r - b) × a = 0

so, r = b + λ a

= (2i - k) + λ(i + j)

= (2 + λ)i + λ j - k.......(1)

similarly, r × b = a × b

⇒(r - a) × b = 0

⇒r = a + η b

= (i + j) + η(2i - k)

= (1 + 2η)i + j - η k ......(2)

from equations (1) and (2),

2 + λ = 1 + 2η

λ = 1 and η = 1 putting these in equation (1) or (2)

so, r = 3i + j - k