CUET PG 2023 Mathematics PYQ — The unit vectors orthogonal to the vector and making equal angles… | Mathem Solvex | Mathem Solvex
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CUET PG 2023 — Mathematics PYQ
CUET PG | Mathematics | 2023
The unit vectors orthogonal to the vector −i^+2j^+2k^ and making equal angles with x and y axis are
Choose the correct answer:
A.
±31(2i^+2j^−k^)
(Correct Answer)
B.
±31(i^+j^−k^)
C.
±31(2i^−2j^−2k^)
D.
±31(i^+2j^−2k^)
Correct Answer:
±31(2i^+2j^−k^)
Explanation
Step 1: Set up the Conditions
Equal angles with x and y axes: If a vector makes equal angles with the x and y axes, their direction cosines must be equal (or the components must be equal).
x=y
So, our vector becomes v=xi^+xj^+zk^.
Orthogonal to −i^+2j^+2k^:
The dot product must be zero.
(xi^+xj^+zk^)⋅(−i^+2j^+2k^)=0
−x+2x+2z=0
x+2z=0⟹x=−2z
Unit Vector Condition:
The magnitude must be equal to 1.
∣v∣=x2+x2+z2=1
2x2+z2=1
Step 2: Solve for the Variables
Substitute x=−2z into the magnitude equation:
2(−2z)2+z2=1
2(4z2)+z2=1
8z2+z2=1
9z2=1⟹z2=91
z=±31
Now, find x and y using x=y=−2z:
If z=31, then x=y=−32
If z=−31, then x=y=32
Final Result
The two possible unit vectors are:
v=±(32i^+32j^−31k^)
Or, written separately:
v1=31(2i^+2j^−k^)
v2=31(−2i^−2j^+k^)
Explanation
Step 1: Set up the Conditions
Equal angles with x and y axes: If a vector makes equal angles with the x and y axes, their direction cosines must be equal (or the components must be equal).
