NIMCET 2023 Mathematics PYQ — If a vector having magnitude of 5 units, makes equal angle with e… | Mathem Solvex | Mathem Solvex
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NIMCET 2023 — Mathematics PYQ
NIMCET | Mathematics | 2023
If a vector having magnitude of 5 units, makes equal angle with each of the three mutually perpendicular axes, then the sum of the magnitude of the projections on each of the axis is
Choose the correct answer:
A.
15/3 unit
B.
53 unit
(Correct Answer)
C.
2153 units
D.
None of these
Correct Answer:
53 unit
Explanation
1. Given Information:
Magnitude of the vector ∣r∣=5.
The vector makes equal angles with the x,y, and z axes. Let these angles be α,β, and γ.
Since α=β=γ, their direction cosines are also equal: l=m=n.
2. Using the Direction Cosine Identity:
We know that:
l2+m2+n2=1
Since l=m=n:
3l2=1⟹l2=31
l=m=n=31
3. Projection on each axis:
The projection of a vector r on an axis is given by the magnitude multiplied by the direction cosine of that axis.
Projection on x-axis (p1) = ∣r∣l=5×31
Projection on y-axis (p2) = ∣r∣m=5×31
Projection on z-axis (p3) = ∣r∣n=5×31
4. Sum of the magnitudes of the projections:
Sum=p1+p2+p3
Sum=35+35+35
Sum=3×35
Sum=315
5. Rationalizing the result:
Sum=3×315×3=3153=53
Correct Option: 2
Explanation
1. Given Information:
Magnitude of the vector ∣r∣=5.
The vector makes equal angles with the x,y, and z axes. Let these angles be α,β, and γ.
Since α=β=γ, their direction cosines are also equal: l=m=n.
2. Using the Direction Cosine Identity:
We know that:
l2+m2+n2=1
Since l=m=n:
3l2=1⟹l2=31
l=m=n=31
3. Projection on each axis:
The projection of a vector r on an axis is given by the magnitude multiplied by the direction cosine of that axis.
