IIT JEE , AIEEE Forum - Mathematics , Vectors

arushsharma

On 21st march you gave a question on vectors which says that a unit vector is orthogonal to 3i+2j+6k and is coplanar with 2i+j+k and i-j+k. So I have taken a unit vector to be (n), which clearly indicates that n.(3i+2j+6k)=0 and also n.(2i+j+k x i-j+k)=0. So we get two equations and if we solve that two equation s we get a unit vector to be 0. So my answer is not matching with your answer. Also instead of writing 3i+2j+6k you have written 5i+2j+6k. Please do help me in this question.

avinash.sharma

arushsharma I am unable to collect the reference of previous question but going to solve the problem again for you

Let the unit vector is A = ai + bj +ck then we have a²+b²+c²=1 .......(1)

Again it is orthogonal to 3i + 2j + 6k so

(ai + bj +ck).( 3i + 2j + 6k) = 0

Þ 3a + 2b + 6c =0 ...... (2)

This unit vector is coplanar with 2i + j + k and i - j + k so

(ai + bj +ck). (2i + j + k x i - j + k) =0 In the determinant form we can write

| a b c |

| 2 1 1 | = 0 Þ a (1+1) -b (2-1) +c(-2-1) = 0

| 1 -1 1 |

Þ 2a - b -3c = 0 ......( 3)

To solve these replace 2b+6c in (2) from (3) as b+3c= 2a so 2b + 6c = 4a. Hence we get a = 0 now b+3c =0 which yields b = -3c Now replace these values in (1) we get

0 + 9 c² + c² =1

Þ 10 c² = 1 Þ c = ± (1/Ö10) and b = -+ (3/Ö10) as b and c have opposite sign

If b = + (3/Ö10) then c = - (1/Ö10) or

if b = - (3/Ö10) then c = + (1/Ö10)

So there two unit vectors + (3/Ö10) j - (1/Ö10) k and

- (3/Ö10) j + (1/Ö10) k these satisfy the given conditions.

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