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MATH 32A Final Exam | Questions and Solutions | 2026 Revised Update | 100% Correct - UCLA.

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SECTION 1: VECTORS AND GEOMETRY IN SPACE (Questions 1-12) Question 1: Given points P(1, -3, -2), Q(2, 0, -4), and R(6, -2, -5), determine whether triangle PQR is right-angled. A. Yes, it is right-angled B. No, it is not right-angled C. Cannot be determined D. It is an equilateral triangle Answer: A. Yes, it is right-angled Rationale: To determine if the triangle is right-angled, compute the vectors representing the sides and check for perpendicularity using the dot product : • PQ = ⟨1, 3, -2⟩ • QR = ⟨4, -2, -1⟩ • PR = ⟨5, 1, -3⟩ PQ · QR = (1)(4) + (3)(-2) + (-2)(-1) = 4 - 6 + 2 = 0. Since the dot product is zero, vectors PQ and QR are perpendicular, making the triangle right-angled at Q.

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MATH 32A Final Exam | Questions and Solutions |
2026 Revised Update | 100% Correct - UCLA.

,
, SECTION 1: VECTORS AND GEOMETRY IN SPACE (Questions 1-12)


Question 1: Given points P(1, -3, -2), Q(2, 0, -4), and R(6, -2, -5), determine
whether triangle PQR is right-angled.
A. Yes, it is right-angled
B. No, it is not right-angled
C. Cannot be determined
D. It is an equilateral triangle
Answer: A. Yes, it is right-angled
Rationale: To determine if the triangle is right-angled, compute the vectors
representing the sides and check for perpendicularity using the dot product :
• PQ = ⟨1, 3, -2⟩
• QR = ⟨4, -2, -1⟩
• PR = ⟨5, 1, -3⟩

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