Question

For any two points \( P, Q \in \mathbb{R}^3 \), let \( \vec{P, Q} \) denote the vector from the point \( P \) to the point \( Q \) and let \( d(P, Q) \) denote the length of \( \vec{P, Q} \). Let \( F: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear transformation such that

• A. \( F \) is an injective map.
• B. \( F \) is a surjective map.
• C. For any four points \( P, Q, R, S \in \mathbb{R}^3 \), we have \[ \vec{(F(P), F(Q))} \cdot \vec{(F(R), F(S))} = \vec{(P, Q)} \cdot \vec{(R, S)}, \] where \( \cdot \) denotes the usual dot product in \( \mathbb{R}^3 \).
• D. For any four points \( P, Q, R, S \in \mathbb{R}^3 \), we have \[ \vec{(F(P), F(Q))} \times \vec{(F(R), F(S))} = \vec{(P, Q)} \times \vec{(R, S)}, \] where \( \times \) denotes the usual cross product in \( \mathbb{R}^3 \).

Hint:

Isometries in \( \mathbb{R}^3 \) preserve lengths, angles, and the cross product, but they are not always surjective, particularly when they involve transformations like reflections or rotations that do not cover the entire space.

Answer 1

The Correct Option is B

Step 1: Analyzing injectivity and surjectivity of \( F \).
Since the transformation \( F \) preserves distances, it is a type of isometry, meaning the transformation does not distort distances. Isometries in \( \mathbb{R}^3 \) can be either translations, rotations, or reflections, which preserve distances but may or may not be injective or surjective. Statement (B) is false because not all isometries are surjective.

Step 2: Check statement (A) on injectivity.
Isometries are injective because if \( F(P) = F(Q) \), then the distances are preserved, implying that \( P = Q \). Therefore, statement (A) is true.

Step 3: Analyze statement (C) involving dot product.
Since \( F \) preserves distances and is a linear transformation, the dot product is preserved. Therefore, statement (C) is true.

Step 4: Analyze statement (D) involving cross product.
The preservation of the cross product is also a property of isometries. The magnitude and direction of the vectors in the cross product are preserved, so statement (D) is true.

Step 5: Conclusion.
From the analysis, we conclude that statement (B) is false because \( F \) may not always be surjective, even though it is an isometry. Therefore, the correct answer is (B).