Pure Mathematics — First: Algebra · Mathematics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 67 questions on this lesson, each with an explanation.
The component of the vector A = 3î − ĵ + 2k̂ in the direction of the vector B = 2î + 2ĵ + k̂ equals ..........
Correct answer: (D) 2
Explanation: Step 1: A · B = (3)(2) + (−1)(2) + (2)(1) = 6. Step 2: ‖B‖ = √(4 + 4 + 1) = 3. Step 3: A_B = (A · B)/‖B‖ = 6/3 = 2. ∴ 2 (6/√14 divides by ‖A‖, which is the component of B on A; 2/3 divides by ‖B‖²).
The vectors a = î + 2ĵ − k̂ and b = 3î − ĵ + k̂ satisfy a · b = 0. This means the two vectors are:
Correct answer: (C) Perpendicular
Explanation: Step 1: a · b = (1)(3) + (2)(−1) + (−1)(1) = 3 − 2 − 1 = 0. Step 2: If the scalar product of two non‑zero vectors is 0, the angle between them is 90°. ∴ The vectors are perpendicular
For a = 2î + ĵ + 2k̂, find a · a.
Correct answer: (A) 9
Explanation: Step 1: a · a = 2² + 1² + 2² = 4 + 1 + 4. Step 2: = 9 (note a · a = ‖a‖²). ∴ a · a = 9
Find the area of the triangle with vertices A(1, 0, 0), B(0, 1, 0), C(0, 0, 1).
Correct answer: (D) √3/2
Explanation: Step 1: AB = (−1, 1, 0), AC = (−1, 0, 1). Step 2: AB × AC = (1·1 − 0·0, 0·(−1) − (−1)·1, (−1)·0 − 1·(−1)) = (1, 1, 1). Step 3: ‖AB × AC‖ = √(1+1+1) = √3. Step 4: Area = ½√3 = √3/2. ∴ The area = √3/2
If A = (2, −1, 3) and B = (1, 2, −1), find (A + B) · (A × B).
Correct answer: (C) 0
Explanation: Step 1: A × B is perpendicular to both A and B, so A · (A × B) = 0 and B · (A × B) = 0. Step 2: (A + B) · (A × B) = A · (A × B) + B · (A × B) = 0 + 0 = 0. Step 3: Check: A × B = (−5, 5, 5) and A + B = (3, 1, 2), so the product = −15 + 5 + 10 = 0. ∴ 0
Find the area of the parallelogram whose adjacent sides are a = 2î and b = 3ĵ.
Correct answer: (B) 6
Explanation: Step 1: a × b = (2î) × (3ĵ) = 6(î × ĵ) = 6k̂. Step 2: Area = ‖a × b‖ = ‖6k̂‖ = 6. ∴ The area = 6
Find a × b where a = î + 2ĵ + 3k̂ and b = 4î + 5ĵ + 6k̂.
Correct answer: (C) −3î + 6ĵ − 3k̂
Explanation: Step 1: a × b = (y₁z₂ − z₁y₂)î − (x₁z₂ − z₁x₂)ĵ + (x₁y₂ − y₁x₂)k̂. Step 2: î: (2)(6) − (3)(5) = 12 − 15 = −3. Step 3: ĵ: −[(1)(6) − (3)(4)] = −(6 − 12) = 6. Step 4: k̂: (1)(5) − (2)(4) = 5 − 8 = −3. ∴ a × b = −3î + 6ĵ − 3k̂
If ‖a‖ = 3, ‖b‖ = 4, ‖a × b‖ = 6√3 and the angle between a and b is acute, find ‖a + b‖.
Correct answer: (A) √37
Explanation: Step 1: ‖a × b‖ = 3 × 4 × sin θ = 6√3 ⇒ sin θ = √3/2, and θ is acute ⇒ θ = 60°, so a · b = 12 cos 60° = 6. Step 2: ‖a + b‖² = ‖a‖² + ‖b‖² + 2a · b = 9 + 16 + 12 = 37 ⇒ ‖a + b‖ = √37 (√13 would be ‖a − b‖). ∴ √37
Find the volume of the parallelepiped with edges a = 2î − ĵ + k̂, b = î + 2ĵ − k̂, c = 3î + k̂.
Correct answer: (A) 2
Explanation: Step 1: Volume = |a · (b × c)| with a = (2,−1,1), b = (1,2,−1), c = (3,0,1). Step 2: b × c = (2·1 − (−1)·0, (−1)·3 − 1·1, 1·0 − 2·3) = (2, −4, −6). Step 3: a · (b × c) = (2)(2) + (−1)(−4) + (1)(−6) = 4 + 4 − 6 = 2. Step 4: Volume = |2| = 2. ∴ The volume = 2
ABCD is a parallelogram with A(1, 2, 3), B(3, −1, 2) and C(4, 1, 0). Find its area.
Correct answer: (D) √122 square units
Explanation: Step 1: AB = (2, −3, −1) and BC = (1, 2, −2) are adjacent sides; AB × BC = (6 + 2, −1 + 4, 4 + 3) = (8, 3, 7). Step 2: Area = ‖AB × BC‖ = √(64 + 9 + 49) = √122; halving it would give the area of triangle ABC only. ∴ √122 square units
The first 10 questions in every subject are free, each with an explanation.