JEE Main Vectors & 3D Geometry: Vector Algebra – Chapter Test

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Difficulty: 10 medium · 10 hard

Vector Algebra MCQ questions with answers

The first 3 of 20 questions. Unlock the series for the rest, each with a solution.

1. Q1. Let a and b be non-zero vectors. Consider the following statements. 1. If a·b > 0, the angle between a and b is acute. 2. If a·b < 0, the angle between a and b is obtuse. 3. If a·b = 0, then at least one of a and b must be the zero vector. Which of these statements is/are correct?

   मान लीजिए a और b शून्येतर सदिश हैं। निम्नलिखित कथनों पर विचार कीजिए। 1. यदि a·b > 0 हो, तो a और b के बीच का कोण न्यून कोण होता है। 2. यदि a·b < 0 हो, तो a और b के बीच का कोण अधिक कोण होता है। 3. यदि a·b = 0 हो, तो a और b में से कम से कम एक शून्य सदिश होना चाहिए। इनमें से कौन-सा/कौन-से कथन सही है/हैं?

   • A. 1 and 2 only / केवल 1 और 2
   • B. 2 and 3 only / केवल 2 और 3
   • C. 1 and 3 only / केवल 1 और 3
   • D. 1, 2 and 3 / 1, 2 और 3
   Solution

   Since a·b = |a||b|cosθ with |a|, |b| > 0, the sign of a·b matches the sign of cosθ, so statements 1 and 2 are true. Statement 3 is false because two non-zero perpendicular vectors also have a·b = 0.

   चूँकि a·b = |a||b|cosθ जहाँ |a|, |b| > 0, a·b का चिह्न cosθ के चिह्न के समान होता है, अतः कथन 1 और 2 सत्य हैं। कथन 3 असत्य है क्योंकि दो शून्येतर लम्बवत् सदिशों के लिए भी a·b = 0 होता है।

2. Q2. Match List I with List II. List I: A. a·a; B. a×a; C. a·(b×c); D. a×(b×c). List II: 1. scalar triple product; 2. |a|²; 3. zero vector; 4. vector triple product. Choose the correct matching.

   सूची I का सूची II से मिलान कीजिए। सूची I: A. a·a; B. a×a; C. a·(b×c); D. a×(b×c)। सूची II: 1. अदिश त्रिक गुणनफल; 2. |a|²; 3. शून्य सदिश; 4. सदिश त्रिक गुणनफल। सही मिलान चुनिए।

   • A. A-2, B-3, C-1, D-4
   • B. A-2, B-1, C-3, D-4
   • C. A-2, B-3, C-4, D-1
   • D. A-3, B-2, C-1, D-4
   Solution

   a·a = |a|² (A-2), a×a = 0 vector (B-3), a·(b×c) is the scalar triple product (C-1), and a×(b×c) is the vector triple product (D-4).

   a·a = |a|² (A-2), a×a = शून्य सदिश (B-3), a·(b×c) अदिश त्रिक गुणनफल है (C-1), तथा a×(b×c) सदिश त्रिक गुणनफल है (D-4)।

3. Q3. The scalar projection of the vector a = 2i + 3j + 2k on the vector b = i + 2j + k is:

   सदिश a = 2i + 3j + 2k का सदिश b = i + 2j + k पर अदिश प्रक्षेप (scalar projection) है:

   • A. 10/3
   • B. 5√6/6
   • C. 5√6/3
   • D. 2√6
   Solution

   Scalar projection = (a·b)/|b| = (2+6+2)/√6 = 10/√6 = 5√6/3.

   अदिश प्रक्षेप = (a·b)/|b| = (2+6+2)/√6 = 10/√6 = 5√6/3।

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