NDA 2026 Mathematics PYQ — Let and . Consider the following statements: I. are orthogonal in… | Mathem Solvex | Mathem Solvex
Tip:A–D to answerE for explanationV for videoS to reveal answer
NDA 2026 — Mathematics PYQ
NDA | Mathematics | 2026
Let a×b=c and b×c=a.
Consider the following statements:
I. a,b,c are orthogonal in pairs.
II. a,b,c are unit vectors.
Which of the statements given above is/are correct?
मान लीजिए a×b=cऔर b×c=aहै।
निम्नलिखित कथनों पर विचार कीजिए :
I. a,b,cयुग्मों में लंबकोणीय हैं ।
II. a,b,cमात्रक सदिश हैं।
उपर्युक्त कथनों में से कौन-सा/कौन-से सही है/हैं ?
Choose the correct answer:
A.
I only
B.
II only
C.
Both I and II
(Correct Answer)
D.
Neither I nor II
Correct Answer:
Both I and II
Explanation
1. Evaluating Statement I (Orthogonality):
By the definition of the vector cross product, the resulting vector is always perpendicular (orthogonal) to both vectors involved.
Given a×b=c, it follows that c⊥a and c⊥b.
Given b×c=a, it follows that a⊥b and a⊥c.
Since a⊥b, b⊥c, and c⊥a, the vectors are mutually perpendicular, meaning they are orthogonal in pairs. Therefore, Statement I is correct.
2. Evaluating Statement II (Magnitude of vectors):
We analyze the magnitudes using the property ∣u×v∣=∣u∣∣v∣sinθ. Since they are orthogonal, θ=90∘ and sin90∘=1.
From a×b=c:
∣c∣=∣a∣∣b∣sin90∘=∣a∣∣b∣
From b×c=a:
∣a∣=∣b∣∣c∣sin90∘=∣b∣∣c∣
Substitute the first equation into the second:
∣a∣=∣b∣(∣a∣∣b∣)=∣a∣∣b∣2
Assuming a=0, we get ∣b∣2=1, which means ∣b∣=1.
Substituting ∣b∣=1 back into our equations:
∣c∣=∣a∣(1)=∣a∣
∣a∣=(1)∣c∣=∣c∣
Thus, ∣a∣=∣c∣. Since ∣b∣=1, and we know b×c=a, the magnitude of a must be 1 because a×b=c and b×c=a implies a system where if one is a unit vector, they all must be unit vectors to satisfy the cross product magnitude relationship ∣a×b∣=∣c∣. Specifically, ∣a∣=∣b∣∣c∣⟹∣a∣=1⋅∣a∣, which is consistent with ∣a∣=1. Thus, Statement II is correct.
Conclusion: Both I and II are correct.
Correct Option:(c)
Explanation
1. Evaluating Statement I (Orthogonality):
By the definition of the vector cross product, the resulting vector is always perpendicular (orthogonal) to both vectors involved.
Given a×b=c, it follows that c⊥a and c⊥b.
Given b×c=a, it follows that a⊥b and a⊥c.
Since a⊥b, b⊥c, and c⊥a, the vectors are mutually perpendicular, meaning they are orthogonal in pairs. Therefore, Statement I is correct.
2. Evaluating Statement II (Magnitude of vectors):
We analyze the magnitudes using the property ∣u×v∣=∣u∣∣v∣sinθ. Since they are orthogonal, θ=90∘ and sin90∘=1.
From a×b=c:
∣c∣=∣a∣∣b∣sin90∘=∣a∣∣b∣
From b×c=a:
∣a∣=∣b∣∣c∣sin90∘=∣b∣∣c∣
Substitute the first equation into the second:
∣a∣=∣b∣(∣a∣∣b∣)=∣a∣∣b∣2
Assuming a=0, we get ∣b∣2=1, which means ∣b∣=1.
Substituting ∣b∣=1 back into our equations:
∣c∣=∣a∣(1)=∣a∣
∣a∣=(1)∣c∣=∣c∣
Thus, ∣a∣=∣c∣. Since ∣b∣=1, and we know b×c=a, the magnitude of a must be 1 because a×b=c and b×c=a implies a system where if one is a unit vector, they all must be unit vectors to satisfy the cross product magnitude relationship ∣a×b∣=∣c∣. Specifically, ∣a∣=∣b∣∣c∣⟹∣a∣=1⋅∣a∣, which is consistent with ∣a∣=1. Thus, Statement II is correct.
