TS EAMCET · Physics · Mathematics in Physics

An ant starts from the origin and crawls $10 cm$ along the $x$ -axis and then $20 cm$ along the $y$ -axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45 °$ with the $x$ -axis and has a magnitude of $\sqrt{2} cm$ is

A $30 cm$
B $30 \sqrt{2} cm$
C $\frac{30}{\sqrt{2}} cm$
D $15 cm$

Verified Solution

Answer & Solution

Correct Answer

(A) $30 cm$

Step-by-step Solution

Detailed explanation

Ant's displacement vector =10i^+20j^=a→ Position vector of the given point =2cos45° i^+2sin45° j^=i^+j^=b→ Therefore, required dot product will be, a→·b→=10i^+20j^·i^+j^=10+20=30 cm

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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An ant starts from the origin and crawls 10 cm along the x-axis and then 20 cm along the y-axis. The dot product of the ant's displacement vector with the position vector of a point that makes 45° with the x-axis and has a magnitude of 2 cm is

Step-by-step Solution

An ant starts from the origin and crawls $10 cm$ along the $x$ -axis and then $20 cm$ along the $y$ -axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45 °$ with the $x$ -axis and has a magnitude of $\sqrt{2} cm$ is