TS EAMCET · Maths · Vector Algebra

If $\bar{O A} = \bar{i} + 2 \bar{j} + 3 \bar{k}$ and $\bar{O B} = 4 \bar{i} + \bar{k}$ are the position vectors of the points $A$ and $B$ , then the position vector of a point on the line passing through $B$ and parallel to the vector $\bar{O A} \times \bar{O B}$ which is at a distance of $\sqrt{189}$ units from $B$ is

A $6 \hat{i} + 11 \hat{j} - 7 \hat{k}$
B $4 \hat{i} + 11 \hat{j} - 8 \hat{k}$
C $2 \hat{i} + 11 \hat{j} - 8 \hat{k}$
D $- 2 \hat{i} - 11 \hat{j} + 8 \hat{k}$

Verified Solution

Answer & Solution

Correct Answer

(A) $6 \hat{i} + 11 \hat{j} - 7 \hat{k}$

Step-by-step Solution

Detailed explanation

Let, C=x,y,z such that BC→ is parallel to OA→×OB→ and BC→=189. Apply BC→=λOA→×OB→ ⇒x-4i^+yj^+z-1k^=λi^j^k^123401 ⇒x-4i^+yj^+z-1k^=λ2i^+11j^-8k^ ⇒x=4+2λ, y=11λ, z=1-8λ…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If OA¯=i¯+2j¯+3k¯ and OB¯=4i¯+k¯ are the position vectors of the points A and B, then the position vector of a point on the line passing through B and parallel to the vector OA¯×OB¯ which is at a distance of 189 units from B is

Step-by-step Solution