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MCQEasyJEE 2024Relative Motion

JEE Physics 2024 Question with Solution

A vector has a magnitude equal to that of A⃗=−3i^+4j^\vec{A} = -3\hat{i} + 4\hat{j} and is parallel to B⃗=4i^+3j^\vec{B} = 4\hat{i} + 3\hat{j}. The xx and yy components of this vector in the first quadrant are xx and yy, respectively. The value of xx is:

  • A

    11

  • B

    22

  • C

    33

  • D

    44

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: A vector has magnitude equal to ∣A⃗∣\left|\vec{A}\right| where A⃗=−3i^+4j^\vec{A} = -3\hat{i} + 4\hat{j}, and it is parallel to B⃗=4i^+3j^\vec{B} = 4\hat{i} + 3\hat{j}.

Find: The xx-component of the required vector in the first quadrant.

First, find the magnitude of A⃗\vec{A}:

∣A⃗∣=(−3)2+42=9+16=5\left|\vec{A}\right| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5

So, the required vector has magnitude 55.

Now find the magnitude of B⃗\vec{B}:

∣B⃗∣=42+32=16+9=5\left|\vec{B}\right| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5

Hence, the unit vector along B⃗\vec{B} is:

B^=B⃗∣B⃗∣=4i^+3j^5\hat{B} = \frac{\vec{B}}{\left|\vec{B}\right|} = \frac{4\hat{i} + 3\hat{j}}{5}

Since the required vector is parallel to B⃗\vec{B} and lies in the first quadrant, its direction is the same as B⃗\vec{B}. Therefore,

V⃗=5B^=5(4i^+3j^5)=4i^+3j^\vec{V} = 5\hat{B} = 5\left(\frac{4\hat{i} + 3\hat{j}}{5}\right) = 4\hat{i} + 3\hat{j}

Thus, the components are x=4x = 4 and y=3y = 3.

Therefore, the correct option is D.

Common mistakes

  • Using the direction of A⃗\vec{A} instead of B⃗\vec{B}. The vector only has the same magnitude as A⃗\vec{A}, but it is parallel to B⃗\vec{B}. Use ∣A⃗∣\left|\vec{A}\right| for magnitude and B⃗\vec{B} for direction.

  • Assuming the required vector is −4i^−3j^-4\hat{i} - 3\hat{j}. That vector is also parallel to B⃗\vec{B} in a broad sense of collinearity, but it is not in the first quadrant. Both components must be positive here.

  • Forgetting to convert B⃗\vec{B} into a unit vector before scaling. If you directly take B⃗\vec{B} as the answer, you may miss the logic that the required vector is magnitude times unit direction. First find B^\hat{B}, then multiply by 55.

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