Question

The scaled and rotated versions of vectors [1, 2] and [-3, 4] are .............

• A. [-1, 3], [-7, 1]
• B. [5, 7], [-7, 3]
• C. [2, -3], [-7, 1]
• D. [2, -3], [-7, 3]

Hint:

For “rotation + uniform scaling’’: (i) angle between vectors is preserved; (ii) the ratio of lengths is unchanged; (iii) dot products scale by the same $s^2$ factor.

Answer 1

The Correct Option is A

A “scaled + rotated’’ transform has the form \( T = sR \), where \( R \) is a \( 2 \times 2 \) rotation matrix and \( s>0 \) is a scalar.
Such a transform preserves the angle between vectors and scales all lengths by the same factor \( s \); hence for input vectors
\( \mathbf{u} = [1, 2] \) and \( \mathbf{v} = [-3, 4] \): \[ \frac{\|\mathbf{v}'\|}{\|\mathbf{u}'\|} = \frac{s\|\mathbf{v}\|}{s\|\mathbf{u}\|} = \frac{\|\mathbf{v}\|}{\|\mathbf{u}\|} = \frac{5}{\sqrt{5}} = \sqrt{5}. \] Check option (A): \( \|\,[\!-1, 3]\,\| = \sqrt{10} \) and \( \|\,[\!-7, 1]\,\| = \sqrt{50} \), so the ratio is \( \sqrt{50}/\sqrt{10} = \sqrt{5} \) — matches.
Also, the dot product must scale by \( s^2 \): \( \mathbf{u} \cdot \mathbf{v} = 5 \). With \( s = \|\,[\!-1, 3]\,\| / \|\mathbf{u}\| = \sqrt{10}/\sqrt{5} = \sqrt{2} \), we get \( s^2(\mathbf{u} \cdot \mathbf{v}) = 2 \times 5 = 10 \), and indeed \([ \!-1, 3 ] \cdot [ \!-7, 1 ] = 7 + 3 = 10 \). Options (B)–(D) fail the constant-length-ratio test, so (A) is the only valid pair.