GATE CSE 2021 Set 2 | Question: 37

Question

For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ is defined as follows:

$$s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])$$

Let $\mathcal{L}$ be a set of $10$-dimensional non-zero real vectors such that for every pair of distinct vectors $P,Q \in  \mathcal{L}$, $s(P,Q)=0$. What is the maximum cardinality possible for the set $\mathcal{L}$?

• A. $9$
• B. $10$
• C. $11$
• D. $100$

Comments

Is this is part of Set Theory bcoz i not studied this type of quesiton in set theory and also concept anyone can please recall me which set theory concept is used here ❓

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This is vector algebra

Answer 1

$S(P, Q)$ is nothing but the dot product of two vectors.

The dot product of two vectors is zero when they are perpendicular, as we are dealing with $10$ dimensional vectors the maximum number of mutually-perpendicular vectors can be $10.$

So option B.

Comments

why cant the vector [0000...0] be included in the set along with other 10 dimensional basis vectors..

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@sukesh_reddy in the question it is given to consider NON-ZERO vectors

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Thank you bro