TS EAMCET · Maths · Three Dimensional Geometry

If $P$ is a point on the line parallel to the vector $2 \hat{i} - 3 \hat{j} - 6 \hat{k}$ and passing through the point A whose position vector is $\hat{i} + 2 \hat{j} - 2 \hat{k}$ and $A P = 21$ , then the position vector of $P$ can be

A $6 \hat{i} - 9 \hat{j} - 18 \hat{k}$
B $6 \hat{k} + 9 \hat{j} - 1 \hat{k}$
C $7 \hat{i} + 11 \hat{j} + 16 \hat{k}$
D $5 \hat{i} - 11 \hat{j} + 16 \hat{k}$

Verified Solution

Answer & Solution

Correct Answer

(C) $7 \hat{i} + 11 \hat{j} + 16 \hat{k}$

Step-by-step Solution

Detailed explanation

Given line passing through A having position vector a→=i^+2j^-2k^ And line parallel to b→=2i^-3j^-6k^ Therefore, equation of line is r→=i^+2j^-2k^+λ2i^-3j^-6k^ ⇒r→=1-2λi^+2-3λj^+-2-6λk^ Let, the position vector of P is…

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 P is a point on the line parallel to the vector 2i^-3j^-6k^ and passing through the point A whose position vector is i^+2j^-2k^ and AP=21, then the position vector of P can be

Step-by-step Solution

If $P$ is a point on the line parallel to the vector $2 \hat{i} - 3 \hat{j} - 6 \hat{k}$ and passing through the point A whose position vector is $\hat{i} + 2 \hat{j} - 2 \hat{k}$ and $A P = 21$ , then the position vector of $P$ can be