GATE DA 2026 | Question: 1

Question

For a classification problem, Principal Component Analysis (PCA) has been used to reduce the dimensionality of a feature space from $100$ to $10$.

Which of the following options is true about the angle $\theta$ between the first and the tenth principal components?

• A. $\theta=0^{\circ}$
• B. $\theta=90^{\circ}$
• C. $90^{\circ}<\theta \leq 180^{\circ}$
• D. $0<\theta<90^{\circ}$

Answer 1

You can see at the corner of your room that there is   : 

• 1 left edge of the floor.

• 1 right edge of the floor.

• And A vertical line going up the wall.

All have 90 degree angle between them. 

 

Every single Principal Component (PC1, PC2, PC10, PC99) acts exactly like  edges in the corner.

That's why answer is 90 Degree. 

Option B = Correct 

Answer 2

In PCA:

1) The first principal component captures the direction of maximum variance.

2) All principal components are mutually orthogonal (i.e., pairwise perpendicular).

Since principal components are orthogonal eigenvectors of the covariance matrix,

\[
v_i^T v_j = 0 \quad \text{for } i \ne j
\]

Hence, the angle between any two distinct principal components:

\[
\cos \theta = 0
\]

Therefore,

\[
\theta = 90^\circ
\]

Correct option:
\[
{(B)}
\]